LOANED   TO 
UNIVERSITY   OF    CALIFORNIA 

DEPARTMENT    OF     MECHANICAL    AND     ELECTRICAL     ENGINEERING 
FROM     PRIVATE    LIBRARY    OF 

C.    L.    CORY 

1930 


I 


APPLIED  T 

'OR  ft 


•YNAMICS 

NEERS  " 


BY 


WILLIAM   D.    ENNIS,   M.E. 

MEMBER    OF    AMERICAN    SOCIETY   OF    MECHANICAL    ENGINEERS 

PROFESSOR    OF    MECHANICAL    ENGINEERING    IN    THE 

POLYTECHNIC    INSTITUTE    OF    BROOKLYN 


WITH  316  ILLUSTRATIONS 


NEW   YOKK 
D.   VAN    NOSTRAND    COMPANY 

23  MURRAY  AND  27  WARREN  STREETS 

1910 


es 


COPYBIGHT,    1910,    BY 

D.  VAN  NOSTRAND  COMPANY 


Norfaiooti  IfJresg 

J.  S.  Gushing  Co. —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


4 


PREFACE 

"  APPLIED  THERMODYNAMICS  "  is  a  pretty  broad  title  ;  but  it  is 
intended  to  describe  a  method  of  treatment  rather  than  unusual 
scope.  The  writer's  aim  has  been  to  present  those  fundamental 
principles  which  concern  the  designer  no  less  than  the  technical 
student  in  such  a  way  as  to  convince  of  their  importance. 

The  vital  problem  of  the  day  in  mechanical  engineering  is  that 
of  the  prime  mover.  Is  the  steam  engine,  the  gas  engine,  or  the 
turbine  to  survive?  The  internal  combustion  engine  works  with 
the  wide  range  of  temperature  shown  by  Carnot  to  be  desirable ; 
but  practically  its  superiority  in  efficiency  is  less  marked  than  its 
temperature  range  should  warrant.  In  most  forms,  its  entire  charge, 
and  in  all  forms,  the  greater  part  of  its  charge,  must  be  compressed 
by  a  separate  and  thermally  wasteful  operation.  By  using  liquid 
or  solid  fuel,  this  complication  may  be  limited  so  as  to  apply  to  the 
air  supply  only ;  but  as  this  air  supply  constitutes  the  greater  part 
of  the  combustible  mixture,  the  difficulties  remain  serious,  and  there 
is  no  present  means  available  for  supplying  oxygen  in  liquid  or  solid 
form  so  as  to  wholly  avoid  the  necessity  for  compression. 

The  turbine,  with  superheat  and  high  vacuum,  has  not  yet 
surpassed  the  best  efficiency  records  of  the  reciprocating  engine, 
although  commercially  its  superior  in  many  applications.  Like  the 
internal  combustion  engine,  the  turbine,  with  its  wide  temperature 
range,  has  gone  far  toward  offsetting  its  low  efficiency  ratio ;  where 
the  temperature  range  has  been  narrow  the  economy  has  been  low, 
and  when  running  non-condensing  the  efficiency  of  the  turbine  has 
compared  unfavorably  with  that  of  the  engine.  There  is  promise 
of  development  along  the  line  of  attack  on  the  energy  losses  in  the 
turbine ;  there  seems  little  to  be  accomplished  in  reducing  these 
losses  in  the  engine.  The  two  motors  may  at  any  moment  reach 
a  parity. 

iii 

798248 


iv  PREFACE 

These  are  the  questions  which  should  be  kept  in  mind  by  the 
reader.  Thermodynamics  is  physics,  not  mathematics  or  logic. 
This  book  takes  a  middle  ground  between  those  text-books  which 
replace  all  theo^  by  empiricism  and  that  other  class  of  treatises 
which  are  too  apt  to  ignore  the  engineering  significance  of  their 
vocabulary  of  differential  equations.  We  here  aim  to  present  ideal 
operations,  to  show  how  they  are  modified  in  practice,  to  amplify 
underlying  principles,  and  to  stop  when  the  further  application  of 
those  principles  becomes  a  matter  of  machine  design.  Thermo- 
dynamics has  its  own  distinct  and  by  no  means  narrow  scope,  and 
the  intellectual  training  arising  from  its  study  is  not  to  be  ignored. 
We  here  deal  only  with  a  few  of  its  engineering  aspects ;  but  these, 
with  all  others,  hark  back  invariably  to  a  few  fundamental  princi- 
ples, and  these  principles  are  the  matters  for  insistent  emphasis. 
Too  much  anxiety  is  sometimes  shown  to  quickly  reach  rules  of 
practice.  This,  perhaps,  has  made  our  subject  too  often  the  barren 
science.  Rules  of  practice  eternally  change ;  for  they  depend  not 
alone  on  underlying  theory,  but  on  conditions  current.  Our  theory 
should  be  so  sound,  and  our  grasp  of  underlying  principles  so  just, 
that  we  may  successfully  attack  new  problems  as  they  arise  and 
evolve  those  rules  of  practice  which  at  any  moment  may  be  best 
for  the  conditions  existing  at  that  moment. 

But  if  Thermodynamics  is  not  differential  equations,  neither 
should  too  much  trouble  be  taken  to  avoid  the  use  of  mathematics 
which  every  engineer  is  supposed  to  have  mastered.  The  calculus 
is  accordingly  employed  where  it  saves  time  and  trouble,  not  else- 
where. The  so-called  general  mathematical  method  has  been  used 
in  the  one  application  where  it  is  still  necessary  ;  elsewhere,  special 
methods,  which  give  more  physical  significance  to  the  things  de- 
scribed, have  been  employed  in  preference.  Formulas  are  useful 
to  the  busy  engineer,  but  destructive  to  the  student;  and  after 
weighing  the  matter  the  writer  has  chosen  to  avoid  formal  definitions 
and  too  binding  symbols,  preferring  to  compel  the  occasionally 
reluctant  reader  to  grub  out  roots  for  himself  —  an  excellent  exer- 
cise which  becomes  play  by  practice. 

The  subject  of  compressed  air  is  perhaps  not  Thermodynamics, 
but  it  illustrates  in  a  simple  way  many  of  the  principles  of  gases 


PREFACE  v 

and  has  therefore  been  included.  Some  other  topics  may  convey 
an  impression  of  novelty ;  the  gas  engine  is  treated  before  the  steam 
engine,  because  if  the  order  is  reversed  the  reader  will  usually  be 
rusty  on  the  theory  of  gases  after  spending  some  weeks  with  vapor 
phenomena ;  a  brief  exposition  of  multiple-effect  distillation  is  pre- 
sented; a  limit  is  suggested  for  the  efficiency  of  the  power  gas 
producer ;  and,  carrying  out  the  general  use  of  the  entropy  diagram 
for  illustrative  purposes,  new  entropy  charts  have  been  prepared 
for  ammonia,  ether,  and  carbon  dioxide.  A  large  number  of  prob- 
lems has  been  incorporated.  Most  of  these  should  be  worked  with 
the  aid  of  the  slide  rule. 

Further  originality  is  not  claimed.  The  subject  has  been  written, 
and  may  now  be  only  re-presented.  All  standard  works  have  been 
consulted,  and  an  effort  has  been  made  to  give  credit  for  methods 
as  well  as  data.  Yet  it  would  be  impossible  in  this  way  to  fully 
acknowledge  the  beneficial  influence  of  the  writer's  former  teachers, 
the  late  Professor  Wood,  Professor  J.  E.  Denton,  and  Dr.  D.  S. 
Jacobus.  It  may  be  sufficient  to  say  that  if  there  is  anything  good 
in  the  book  they  have  contributed  to  it ;  and  for  what  is  not  good, 
they  are  not  responsible. 

POLYTECHNIC  INSTITUTE  OF  BROOKLYN, 
NEW  YORK,  August,  1910. 


CONTENTS 

CHAPTER  -  PAGE 

I.     THE  NATURE  AND  EFFECTS  OF  HEAT .        .        1 

II.     THE    HEAT    UNIT  :    SPECIFIC    HEAT  :    FIRST    LAW   OF    THERMO- 

*     DYNAMICS     ...        . 11 

III.  LAWS  OF  GASES  :   ABSOLUTE  TEMPERATURE  :   THE  PERFECT  GAS        19 

IV.  THERMAL  CAPACITIES  :  SPECIFIC  HEATS  OF  GASES  :  JOULE'S  LAW      29 

V.     GRAPHICAL  REPRESENTATIONS  :  PRESSURE-VOLUME  PATHS  OF  PER- 
FECT GASES .        .39 

VI.     THE  CARNOT  CYCLE     ..........      63 

VII.     THE  SECOND  LAW  OF  THERMODYNAMICS 70 

VIII.     ENTROPY        .        , 77 

IX.     COMPRESSED  AIR 90 

The  cold  air  engine  :  cycle,  temperature  fall,  preheaters,  design  of 
engine :  the  compressor :  cycle,  form  of  compression  curve, 
jackets,  multi-stage  compression,  intercooling,  power  consump- 
tion: engine  and  compressor  relations:  losses,  efficiencies,  en- 
tropy diagram,  compressor  capacity,  volumetric  efficiency, 
design  of  compressor,  commercial  types :  compressed  air  trans- 
mission. 

X.     HOT-AIR  ENGINES .        .        .     129 

XI.     GAS  POWER 146 

The  producer :  limit  of  efficiency :  gas  engine  cycles :  Otto,  Car- 
not,  Atkinson,  Lenoir,  Bray  ton,  Clerk,  Diesel:  practical  modi- 
fications of  the  Otto  cycle:  mixture,  compression,  ignition, 
dissociation,  clearance,  expansion,  scavenging,  diagram  factor: 
principles  of  design  and  efficiency  :  commercial  gas  engines : 
results  of  tests :  gas  engine  regulation. 

XII.     THEORY  OF  VAPORS     .        .        .        .     ' 199 

Formation  at  constant  pressure  :  saturated  steam :  superheated 
steam  :  paths  of  vapors  :  vapors  in  general :  steam  cycles :  steam 
tables. 

vii 


viii  CONTENTS 

CHAPTER  PAGE 

XIII.     THE  STEAM  ENGINE .        .    256 

Practical  modifications  of  the  Raiikine  cycle:  complete  and  in- 
complete expansion,  wiredrawing,  cylinder  condensation,  ratio 
of  expansion,  the  steam  jacket,  use  of  superheated  steam,  actual 
expansion  curve,  mean  effective  pressure,  back  pressure,  clear- 
ance, compression,  valve  action  :  the  entropy  diagram  :  cylinder 
feed  and  cushion  steam,  Boulvin's  method,  Reeve's  method : 
multiple  expansion  :  desirability  of  complete  expansion,  conden- 
sation losses  in  compound  cylinders,  Woolf  engine,  receiver 
engine,  tandem  and  cross  compounds,  combined  diagrams,  de- 
sign of  compound  engines,  governing,  the  drop  controversy, 
intermediate  compounds,  binary  vapor  engine:  engine  tests: 
indicators,  calorimeters,  heat  supplied,  heat  rejected,  heat  trans- 
fers :  types  of  steam  engine. 

XIV.    THE  STEAM  TURBINE        .        ...        .        .        .        .        .    317 

Conversion  of  heat  into  velocity  :  the  turbine  cycle,  effects  of  fric- 
tion, rate  of  flow  :  efficiency  in  directing  velocities :  velocity  com- 
pounding, pressure  compounding:  efficiency  of  the  turbine:  de- 
sign of  impulse  and  pressure  turbines :  commercial  types  and 
applications. 

XV.     RESULTS  OF  TRIALS  OF  STEAM  ENGINES  AND  STEAM  TURBINES     350 
Economy,  condensing  and  non-condensing,  of  various  commercial 
forms  with  saturated  and  superheated  steam :  mechanical  effi- 
ciencies. 

XVI.     THE  STEAM  POWER  PLANT       .  '      .        .        .  .        .        .    360 

Fuels,  combustion  economy,  air  supply,  boilers,  theory  of  draft, 
fans,  chimneys,  stokers,  heaters,  superheaters,  economizers,  con- 
densers, pumps,  injectors. 

XVII.     DISTILLATION      .        .  I V.  .  .        .        ...        .,'      .     380 

The  still,  evaporation  in  vacuo,  multiple-effect  evaporation. 
FUSION  : 

Change  of  volume  during  change  of  state,  pressure-temperature 
relation,  latent  heat  of  fusion  of  ice. 

LIQUEFACTION  OF  GASES: 

Pressure  and  cooling,  critical  temperature,  cascade  system,  regen- 
erative apparatus. 

XVIII.    MECHANICAL  REFRIGERATION •  '•..«.       •    394 

Air  machines:  reversed  cycle,  Bell-Coleman  machine,  dense  air 
apparatus,  coefficient  of  performance,  Kelvin  warming  ma- 
chine: vapor-compression  machines:  the  cycle,  choice  of  fluid, 
tonnage  rating,  ice-melting  effect,  design  of  compressor :  the 
absorption  system  :  methods  and  fields  of  application  :  ice-mak- 
ing: commercial  efficiencies. 


CHAPTER  I 

THE  NATURE  AND  EFFECTS  OF  HEAT: 


1.  Heat  as  Motive  Power.     All  artificial   motive  powers  derive   their 
origin  from  heat.     Muscular  effort,  the  forces  of  the  waterfall,  the  wind, 
tides  and  waves,  and  the  energy  developed  by  the  combustion  of  fuel,  may 
all  be  traced  back  to  reactions  induced  by  heat.     Our  solid,  liquid,  and 
gaseous  fuels  are  stored-up  solar  heat  in  the  forms  of  hydrogen  and  carbon. 

2.  Nature  of  Heat.     We  speak  of  bodies  as  "  hot "  or  "  cold,"  referring 
to  certain   impressions  which   they  produce  upon  our  senses.     Common 
experimental  knowledge  regarding  heat  is  limited  to  sensations  of  temper- 
ature.    Is   heat  matter,  force,  motion,  or  position  ?     The  old  "  caloric " 
theory  was  that  "heat  was  that  substance  whose  entrance  into  our  bodies 
causes  the  sensation  of  warmth,  and  whose  egress  the  sensation  of  cold." 
But  heat  is  not  a  "  substance  "  similar  to  those  with  which  we  are  familiar, 
for  a  hot   body  weighs  no  more  than  one  which  is  cold.     The  calorists 
avoided  this  difficulty  by  assuming  the  existence  of  a  weightless  material 
fluid,  caloric.     This  substance,  present  in  the  interstices  of  bodies,  it  was 
contended,  produced  the  effects  of  heat;  it  had  the  property  of  passing 
between  bodies  over  any  intervening  distance.     Friction,  for  example,  de- 
creased  the  capacity  for  caloric;    and   consequently  some   of  the  latter 
"  flowed  out,"  as  to  the  hand  of  the  observer,  producing  the  sensation  of 
heat.     Davy,  however,  in  1799,  proved  that  friction  does  not  diminish  the 
capacity  of  bodies  for  containing  heat,  by  rubbing  together  two  pieces  of 
ice  until  they  melted.     According  to  the  caloric  theory,  the  resulting  water 
should  have  had  less  capacity  for  heat  than  the  original  ice :  but  the  fact  is 
that  water  has  actually  about  twice  the  capacity  for  heat  that  ice  has ;  or, 
in  other  words,  the  specific  heat  of  water  is  about  1.0,  while  that  of  ice  is 
0.504.     The  caloric  theory  was  further  assailed  by  Eumford,  who  showed 
that  the  supply  of  heat  from  a  body  put  under  appropriate  conditions  was 
so  nearly  inexhaustible  that  the  source  thereof  could  not  be  conceived  as 
being  even    an  "  imponderable"  substance.     The  notion  of  the  calorists 
was  that  the  different  specific  heats  of  bodies  were  due  to  a  varying  capac- 
ity for  caloric ;  that  caloric  might  be  squeezed  out  of  a  body  like  water 
from  a  sponge.     Rumford  measured  the  heat  generated  by  the  boring  of 
cannon  in  the  arsenal  at   Munich.     In  one  experiment,  a  gun  weighing 

1 


APPLIED  THERMODYNAMICS 


11313  Ib.  was  heated  70°  F.,  although  the  total  weight  of  borings  produced 
was  only  837  grains  troy.  In  a  later  experiment,  Rumford  succeeded  in 
boiling  water  by  the  heat  thus  generated.  He  argued  that  "anything 
which  any  insulated  body  or  system  of  bodies  may  continue  to  famish  without 
limitation  cannot  possibly  be  a  material  substance"  The  evolution  of  heat, 

:  it  was   contended, :  might  continue  as  indefinitely  as  the  generation  of 

:  sftiind  following  the  repeated  striking  of  a  bell  (1). 

:  JoUle,  ai>out  1845,  showed  conclusively  that  mechanical  energy 
alone  sufficed  for  the  production  of  heat,  and  that  the  amount  of  heat 

generated  was  always  proportionate  to  the 
energy  expended.  A  view  of  his  apparatus 
is  given  in  Fig.  1,  v  and  h  being  the  verti- 
cal and  horizontal  sections,  respectively,  of 
the  container  shown  at  c.  Water  being 
placed  in  e,  a  rotary  motion  of  the  contained 
brass  paddle  wheel  was  caused  by  the  de- 
scent of  two  leaden  weights  suspended  by 
cords.  The  rise  in  tempeuature  of  the 


va) 


FIG.  1.    Arts.  2,  30.— Joule's  Apparatus. 

water  was  noted,  the  expended  work  (by  the  falling  weights)  com- 
puted, and  a  proper  correction  made  for  radiation.  Similar  experi- 
ments were  made  with  mercury  instead  of  water.  As  a  result  of 
his  experiments,  Joule  reached  conclusions  which  served  to  finally 
overthrow  the  caloric  theory. 

3.  Mechanical  Theory  of  Heat.  Various  ancient  and  modern 
philosophers  had  conceded  that  heat  was  a  motion  of  the  minute 
particles  of  the  body,  some  of  them  suggesting  that  such  motion 


THE  NATURE  AND  EFFECTS   OF  HEAT  3 

was  produced  by  an  "igneous  matter."  Locke  denned  heat  as  "a 
very  brisk  agitation  of  the  insensible  parts  of  the  object,  which  pro- 
duces in  us  that  sensation  from  which  we  denominate  the  object 
hot ;  so  [that]  what  in  our  sensation  is  heat,  in  the  object  is  nothing 
but  motion."  Young  argued,  "If  heat  be  not  a  substance,  it  must 
be  a  quality;  and  this  quality  can  only  be  a  motion."  This  is  the 
modern  conception.  Heat  is  energy :  it  can  perform  work,  or  pro- 
duce certain  sensations ;  it  can  be  measured  by  its  various  effects. 
It  is  regarded  as  "  energy  stored  in  a  substance  by  virtue  of  the  state 
of  its  molecular  motion''  (2). 

Conceding  that  heat  is  energy,  and  remembering  the  expression  for  energy, 
\  ?ny2,  it  follows  that  if  the  mass  of  the  particle  does  not  change,  its  velocity  (molec- 
ular velocity)  must  change ;  or  if  heat  is  to  include  potential  energy,  then  the 
molecular  configuratiofi  must  change.  The  molecular  vibrations  are  invisible,  and 
their  precise  nature  unknown.  Rankine's  theory  of  molecular  vortices  assumes  a 
law  of  vibration  which  has  led  to  some  useful  results. 

Since  heat  is  energy,  its  laws  are  those  generally  applicable  to  energy, 
as  laid  down  by  Newton :  it  must  have  a  commensurable  value ;  it  must 
be  convertible  into  other  forms  of  energy,  and  they  to  heat;  and  the 
equivalent  of  heat  energy,  expressed  in  mechanical  energy  units,  must  be 
constant  and  determinable  by  experiment. 

4.  Subdivisions  of  the  Subject.     The  evolutions  and  absorptions 
of  heat  accompanying  atomic  combinations  and  molecular  decompo- 
sitions are  the  subjects  of  thermochemistry.     The  mutual  relations  of 
heat  phenomena,  with  the  consideration  of  the  laws  of  heat  trans- 
mission, are  dealt  with  in  general  physics.     The  relations  between 
heat  and  mechanical  energy  are  included  in  the  scope  of  applied  engi- 
neering thermodynamics,  which  may  be  denned  as  the  science  of  the 
mechanical  theory  of  heat.    While  thermodynamics  is  thus  apparently 
only  a  subdivision  of  that  branch  of  physics  which  treats  of  heat,  the 
relations  which  it  considers  are  so  important  that  it  may  be  regarded 
as  one  of  the  two  fundamental  divisions  of  physics,  which  from  this 
standpoint   includes   mechanics  —  dealing  with   the   phenomena   of 
ordinary  masses  —  and  thermodynamics  —  treating  of  the  phenomena 
of  molecules. 

5.  Applications    of    Thermodynamics.     The    subject   has    far-reaching 
applications  in  physics  and  chemistry.     In  its  mechanical  aspects,  it  deals 


4  APPLIED  THERMODYNAMICS 

with  matters  fundamental  to  the  engineer.  After  developing  the  general 
laws  and  dwelling  briefly  upon  ideal  processes,  we  are  to  study  the  condi- 
tions affecting  the  efficiency  and  capacity  of  air,  gas,  and  steam  engines 
and  the  steam  turbine;  together  with  the  economics  of  air  compression, 
distillation,  refrigeration,  and  gaseous  liquefaction.  The  ultimate  engi- 
neering application  of  thermodynamics  is  in  the  saving  of  heat,  an  appli- 
cation which  becomes  attractive  when  viewed  in  its  just  aspect  as  a  saving 
of  money  and  a  mode  of  conservation  of  our  material  wealth. 

6.  Temperature.  A  hot  body,  in  common  language,  is  one  whose 
temperature  is  high,  while  a  cold  body  is  one  low  in  temperature.  Tem- 
perature, then,  is  a  measure  of  the  hotness  of  bodies.  From  a  rise  in  tem- 
perature, we  infer  an  accession  of  heat;  or  from  a  fall  in  temperature, 
a  loss  of  heat.  Temperature  is  not,  however,  a  satisfactory  measure  of 
quantities  of  heat.  A  pound  of  water  at  200°  contains  very  much  more 
heat  than  a  pound  of  lead  at  the  same  temperature ;  this  may  be  demon- 
strated by  successively  cooling  the  bodies  in  a  bath  to  the  same  final  tem- 
perature, and  noting  the  gain  of  heat  by  the  bath.  Furthermore,  immense 
quantities  of  heat  are  absorbed  by  bodies  in  passing  from  the  solid  to  the 
liquid  or  from  the  liquid  to  the  vaporous  conditions,  without  any  change 
in  temperature  whatever.  Temperature  defines  a  condition  of  heat  only. 
It  is  a  measure  of  the  capacity  of  the  body  for  communicating  heat  to  other 
bodies.  Heat  always  passes  from  a  body  of  relatively  high  temperature ; 
it  never  passes  of  itself  from  a  cold  body  to  a  hot  one.  Wherever  two 
bodies  of  different  temperatures  are  in  thermal  juxtaposition,  an  inter- 
change of  heat  takes  place ;  the  cooler  body  absorbs  heat  from  the  hotter 
body,  no  matter  which  contains  initially  the  greater  quantity  of  heat, 
until  the  two  are  at  the  same  temperature,  or  in  thermal  equilibrium. 
Two  bodies  are  at  the  same  temperature  when  there  is  no  tendency  toward  a 
transfer  of  heat  between  them.  Measurements  of  temperature  are  in  gen- 
eral based  upon  arbitrary  scales,  standardized  by  comparison  with  some 
physically  established  "  fixed  "  point.  One  of  these  fixed  temperatures  is 
that  minimum  at  which  pure  water  boils  when  under  normal  atmospheric 
pressure  of  14.697  Ib.  per  square  inch;  viz.  212°  F.  Another  is  the 
maximum  temperature  of  melting  ice  at  atmospheric  pressure,  which  is 
32°  F.  Our  arbitrary  scales  of  temperature  cannot  be  expressed  in  terms 
of  the  fundamental  physical  units  of  length  and  weight. 

7-  Measurement  of  Temperature.  Temperatures  are  measured  by  thermome- 
ters. The  common  type  of  instrument  consists  of  a  connected  bulb  and  vertical 
tube,  of  glass,  in  which  are  contained  a  liquid.  Any  change  in  temperature  affects 
the  volume  of  the  liquid,  and  the  portion  in  the  tube  consequently  rises  or  falls. 
The  expansion  of  solids  or  of  gases  is  sometimes  utilized  in  the  design  of  thermom- 


THE  NATURE  AND  EFFECTS   OF  HEAT  5 

eters.  Mercury  and  alcohol  are  the  liquids  commonly  used.  The  former  freezes  at 
-  38°  F.  and  boils  at  675°  F.  The  latter  freezes  at  -  203°  F.  and  boils  at  173°  F. 
The  mercury  thermometer  is,  therefore,  more  commonly  used  for  high  tempera- 
tures, and  the  alcohol  for  low. 

8.  Thermometric  Scales.    The  Fahrenheit  thermometer,  generally 
employed  by  engineers   in   the  United  States   and  Great  Britain, 
divides  the  space  between  the  "fixed  points"  (Art.  6)  into  180 
equal   degrees,  freezing  being  at  32°  and   boiling  at  212°.       The 
Centigrade  scale,  employed  by  chemists  and  physicists  (sometimes 
described  as  the  Celsius  scale),  calls  the  freezing  point  0°  and  the 
boiling  point  100°.     On  the  Reaumur  scale,  used  in  Russia  and  a 
few  other  countries,  water  freezes  at  0°  and  boils  at  80°.     One  de- 
gree on  the  Fahrenheit  scale  is,  therefore,  equal  to  J°  C.,  or  to  |°  R. 
In  making  transformations,  care  must  be  taken  to  regard  the  differ- 
ent zero  point  of  the  Fahrenheit  thermometer.     On  all  scales,  tem- 
peratures below  zero  are  distinguished  by  the  minus  (  — )  prefix. 

The  Centigrade  scale  is  unquestionably  superior  in  facilitating  arithmetical 
calculations;  but  as  most  English  papers  and  tables  are  published  in  Fahrenheit 
units,  we  must,  for  the  present  at  least,  use  that  scale  of  temperatures. 

9.  High  Temperature   Measurements.      For    measuring    temperatures    above 
800°  F.,  some  form  of  pyrometer  must  be  employed.     The  simplest  of  these  is  the 
metallic  pyrometer,  exemplifying  the  principle  that  different  metals  expand  to  dif- 
ferent extents  when  heated  through  the  same  range  of  temperature.     Bars  of  iron 
and  brass  are  firmly  connected  at  one  end,  the  other  ends  being  free.     At  some 
standard  temperature  the  two  bars  are  of  the  same  length,  and  the  indicator,  con- 
trolled jointly  by  the  two  free  ends  of  the  bars,  registers  that  temperature.    When 
the  temperature  changes,  the  indicator  is  moved  to  a  new  position  by  the  relative 
distortion  of  the  free  ends. 

In  the  Le  Chatelier  electric  pyrometer,  a  thermoelectric  couple  is  employed.  For 
temperatures  ranging  from  300°  C.  to  1500°  C.,  one  element  is  made  of  platinum, 
the  other  of  a  10  per  cent,  alloy  of  platinum  with  rhodium.  Any  rise  in  tempera- 
ture at  the  junction  of  the  elements  induces  a  flow  of  electric  current,  which  is  con- 
ducted by  wires  to  a  galvanometer,  located  in  any  convenient  position.  The  ex- 
pensive metallic  elements  are  protected  from  oxidation  by  enclosing  porcelain 
tubes.  In  the  Bristol  thermoelectric  instrument,  one  element  is  of  a  platinum- 
rhodium  alloy,  the  other  of  a  cheaper  metal.  The  electromotive  force  is  indicated 
by  a  Weston  millivoltmeter,  graduated  to  read  temperatures  directly.  The  in- 
strument is  accurate  up  to  2000°  F.  The  electrical  resistance  pyrometer  is  based  on 
the  law  of  increase  of  electrical  resistance  with  increase  of  temperature.  In  Cal- 
lendar's  form,  a  coil  of  fine  platinum  wire  is  wound  on  a  serrated  mica  frame. 
The  instrument  is  enclosed  in  porcelain,  and  placed  in  the  space  the  temperature 


6  APPLIED  THERMODYNAMICS 

of  which  is  to  be  ascertained.  The  resistance  is  measured  by  a  Wheatstone  bridge, 
a  galvanometer,  or  a  potentiometer,  calibrated  to  read  temperatures  directly. 
Each  instrument  must  be  separately  calibrated. 

Optical  pyrometers  are  based  on  the  principle  that  the  colors  of  bodies  vary 
with  their  temperatures.  In  the  Morse  thermogage,  of  this  type,  an  incandescent 
lamp  is  wired  in  circuit  with  a  rheostat  and  a  millivoltmeter.  The  lamp  is  located 
between  the  eye  and  the  object,  and  the  current  is  regulated  until  the  lamp  be- 
comes invisible.  The  temperature  is  then  read  directly  from  the  calibrated  milli- 
voltmeter. The  device  is  extensively  used  in  hardening  steel  tools,  and  has  been 
employed  to  measure  the  temperatures  in  steam  boiler  furnaces. 

10.  Cardinal  Properties.  A  cardinal  or  integral  property  of  a 
substance  is  any  property  which  is  fully  defined  by  the  immediate 
state  of  the  substance.  Thus,  weight,  length,  specific  gravity,  are 
cardinal  properties.  On  the  other  hand,  cost  is  a  non-cardinal  prop- 
erty ;  the  cost  of  a  substance  cannot  be  determined  by  examination 
of  that  substance;  it  depends  upon  the  previous  history  of  the  sub- 
stance. Any  two  or  three  cardinal  properties  of  a  substance  may  be 
used  as  coordinates  in  a  graphic  representation  of  the  state  of  the  sub- 
stance. Properties  not  cardinal  may  not  be  so  used,  because  such 
properties  do  not  determine,  nor  are  they  determinable  by,  the  pres- 
ent state  of  the  substance.  The  cardinal  properties  employed  in 
thermodynamics  are  five  or  six  in  number.*  Three  of  these  are  pres- 
sure, volume,  and  temperature ;  pressure  being  understood  to  mean 
specific  pressure,  or  uniform  pressure  per  unit  of  surface,  exerted  by  or 
upon  the  body,  and  volume  to  mean  volume  per  unit  of  weight.  The 
location  of  any  point  in  space  is  fully  determined  by  its  three  coordi- 
nates. Similarly,  any  three  cardinal  properties  may  serve  to  fix  the 
thermal  condition  of  a  substance. 

The  first  general  principle  of  thermodynamics  is  that  if  two  of  the 
three  named  cardinal  properties  are  known,  these  two  enable  us  to  calcu- 
late the  third.  This  principle  cannot  be  proved  a  priori ;  it  is  to  be  justi- 
fied by  its'  results  in  practice.  Other  thermodynamic  properties  than 
pressure,  volume,  and  temperature  conform  to  the  same  general  principle 
(Art.  169)  ;  with  these  properties  we  are  as  yet  unacquainted.  A  correlated 
principle  is,  then,  that  any  two  of  the  cardinal  properties  suffice  to  fully 
determine  the  state  of  the  substance.  For  certain  gases,  the  general  prin- 
ciple may  be  expressed,  PV=  (  f}T 

*  For  gases,  pressure,  volume,  temperature,  internal  energy,  entropy ;  for  wet 
vapors,  these  five  and  dryness. 


THE  NATURE  AND  EFFECTS   OF  HEAT  7 

while  for  other  gaseous  fluids  more  complex  equations  (Art.  363)  must  be 
used.  In  general,  these  equations  are,  in  the  language  of  analytical  geom- 
etry, equations  to  a  surface.  Certain  vapors  cannot  be  represented,  as 
yet,  by  any  single  equation  between  P,  V,  and  T,  although  corresponding 
values  of  these  properties  may  have  been  ascertained  by  experiment. 

11.  Preliminary  Assumptions.     The  greater  part  of  the  subject 
deals  with  substances  assumed  to  be  in  a  state  of  mechanical  equilibrium, 
all  changes  being  made  with  infinite  slowness.     A  second  assumption 
is  that  no  chemical  actions  occur  during  the  thermodynamic  trans- 
formation.    In  the  third  place,  the  substances  dealt  with  are  assumed 
to  be  so  homogeneous,  as  to  be  in  uniform  thermal  condition  through- 
out :  for  example,  the  pressure  property  must  involve  equality  of 
pressure  in  all  directions ;  and  this  limits  the  consideration  to  the 
properties  of  liquids  and  gases. 

The  thermodynamics  of  solids  is  extremely  complex,  because  of  the  obscure 
stresses  accompanying  their  deformation  (3).  Kelvin  (4)  has  presented  a  general 
analysis  of  the  action  of  any  homogeneous  solid  body  homogeneously  strained. 

12.  The  Three   Effects  of  Heat.     Setting  aside  the  obvious  un- 
classified changes  in  pressure,  volume,  and  temperature  accompanying 
manifestations  of  heat  energy,  there  are  three  known  ways  in  which 
heat  may  be  expended.     They  are  : 

(a)  In  a  change  of  temperature  of  the  substance. 

(5)  In  a  change  of  physical  state  of  the  substance. 

(c)  In  the  performance  of  external  work  by  or  upon  the  substance. 
Denoting  these  effects  by  T,  I,  and  W,  then,  for  any  transfer  of  heat 
H,  we  have  the  relation 

H=T  +  I  +  W, 

any  of  the  terms  of  which  expression  may  be  negative.  It  should  be 
quite  obvious,  therefore,  that  changes  of  temperature  alone  are  in- 
sufficient to  measure  expenditures  of  heat. 

Items  (a)  and  (6)  are  sometimes  grouped  together  as  indications 
of  a  change  in  the  INTERNAL  ENERGY  of  the  heated  substance,  the 
term  being  one  of  the  first  importance,  which  it  is  essential  to  clearly 
apprehend.  Items  (5)  and  (c)  are  similarly  sometimes  combined  as 
representing  the  total  work. 


8  APPLIED  THERMODYNAMICS 

13.  The  Temperature  Effect.     Temperature  indications  of   heat  activity    are 
sometimes  referred  to  as  "  sensible  heat."     The  addition  of  heat  to  a  substance 
may  either  raise  or  lower  its  temperature,  in  accordance  with  the  fundamental 
equation  of  Art.  12. 

The  temperature  effect  of  heat,  from  the  standpoint  of  the  mechanical 
theory,  is  due  to  a  change  in  the  velocity  of  molecular  motion,  in  conse- 
quence of  which  the  kinetic  energy  of  that  motion  changes. 

This  effect  is  therefore  sometimes  referred  to  as  vibration  work.  Clausius 
called  it  actual  energy. 

14.  External  Work  Effect.     The  expansion  of  solids  and  fluids,  due  to  the  supply 
of  heat,  is  a  familiar  phenomenon.     Heat  may  cause  either  expansion  or  contraction, 
which,  if  exerted  against  a  resistance,  may  suffice  to  perform  mechanical  work. 

15.  Changes  of  Physical  State.     Broadly  speaking,  such  effects 
include  all  changes,  other  than  those  of  temperature,  within  the  sub- 
stance itself.     The  most  familiar  examples  are  the  change  between 
the   solid   and   the  liquid  condition,    when  the  substance  melts  or 
freezes,  and  that  between  the  liquid  and  the  vaporous,  when  it  boils 
or  condenses ;  but  there  are  intermediate  changes  of  molecular  aggrega- 
tion in  all  material  bodies  which  are  to  be  classed  with  these  effects 
under  the  general  description,  disgregation  work.     The  mechanical 
theory  assumes  that  in  such  changes  the  molecules  are  moved  into 
new  positions,  with  or  against  the  lines  of  mutual  attraction.     These 
movements  are  analogous  to  the  "partial  raising  or  lowering  of  a 
weight  which  is  later  to  be  caused  to  perform  work  by  its  ovyn  descent. 
The  potential  energy  of  the  substance  is  thus  changed,  and  positive 
or  negative  work  is  performed  against  internal  resisting  forces." 

When  a  substance  changes  its  physical  state,  as  from  water  to  steam,  it 
can  be  shown  that  a  very  considerable  amount  of  external  work  is  done,  in 
consequence  of  the  increase  in  volume  which  occurs,  and  which  may  be 
made  to  occur  against  a  heavy  pressure.  This  external  work  is,  however, 
equivalent  only  to  a  very  small  proportion  of  the  total  heat  supplied  to 
produce  evaporation,  the  balance  of  the  heat  having  been  expended  in  the 
performance  of  disgregation  work. 

The  molecular  displacements  constituting  disgregation  work  are  exemplified  in 
the  phenomena  of  solution,  and  in  the  action  of  freezing  mixtures  (5). 

16.  Solid,  Liquid,  Vapor,  Gas.     Solid  bodies  are  those  which  resist  tendencies 
to  change  their  form  or  volume.     Liquids  are  those  bodies  which  in  all  of  their 


THE  NATURE  AND   EFFECTS   OF  HEAT  9 

parts  tend  to  preserve  definite  volume,  and  which  are  practically  unresistant  to 
influences  tending  to  slowly  change  their  figure.  Gases  are  unresistant  to  slow 
changes  in  figure  or  to  increases  in  volume.  They  tend  to  expand  indefinitely  so 
as  to  completely  fill  any  space  in  which  they  are  contained,  no  matter  what  the 
shape  or  the  size  of  that  space  may  be.  Most  substances  have  been  observed  in 
all  three  forms,  under  appropriate  conditions ;  and  all  substances  can  exist  in  any 
of  the  forms.  At  this  stage  of  the  discussion,  no  essential  difference  need  be 
dra\vn  between  a  vapor  and  a  gas.  Formerly,  the  name  vapor  was  applied  to 
those  gaseous  substances  which  at  ordinary  temperatures  were  liquid,  while  a 
ugas"  was  a  substance  never  observed  in  the  liquid  condition.  Since  all  of  the 
so-called  "  permanent"  gases  have  been  liquefied,  this  distinction  has  lost  its  force. 
A  useful  definition  of  a  vapor  as  distinct  from  a  true  gas  will  be  given  later 
(Art.  380). 

Under  normal  atmospheric  pressure,  there  exist  well-defined  tempera- 
tures at  which  various  substances  pass  from  the  solid  to  the  liquid  and 
from  the  liquid  to  the  gaseous  conditions.  The  temperature  at  which  the 
former  change  occurs  is  called  the  melting  point  or  freezing  point;  that  of 
the  latter  is  known  as  the  boiling  point  or  temperature  of  condensation. 

17.  Other  Changes  of  State.     Although  the  operation  described  as  boiling 
occurs,  for  each  liquid,  at  some  definite  temperature,  there  is  an  almost  continual 
evolution  of  vapor  from  nearly  all  liquids  at  temperatures  below  their  boiling 
points.     The  water  of  the  earth's  surface,  for  example,  is  slowly  changing  to  vapor 
and  impregnating  the  atmosphere.     Such  "insensible"  evaporation  is  with  some 
substances  non-existent,  or  at  least  too  small  in  amount  to  permit  of  measure- 
ment :  as  in  the  instances  of  mercury  at  32°  F.  or  of  sulphuric  acid  at  any  ordi- 
nary temperature.      Ordinarily,  a   liquid   at   a  given   temperature   continues  to 
evaporate  so  long  as  its  partial  vapor  pressure  is  less  than  the  maximum  pressure 
corresponding  to  its  temperature.     The  interesting  phenomenon  of   sublimation 
consists  in  the  direct  passage  from  the  solid  to  the  gaseous  state.     Such  sub- 
stances as  camphor  and  iodine  manifest  this  property.     Ice  and  snow  also  pass 
directly  to  a  state  of  vapor  at  temperatures  far  below  the  freezing  point.     There 
seem  to  be  no  quantitative  data  on  the  heat  relations  accompanying  this  change 
of  state. 

18.  Variations  in   "Fixed  Points."     Aside   from   the   influence   of  pressure 
(Arts.  358,  603),  various  causes  may  modify  the  positions  of  the  "fixed  points  "  of 
the  thermometric  scale.     Water  may  be  cooled  below  32°  F.  without  freezing, 
if  kept  perfectly  still.     If  free  from  air,  water  boils  at  270-290°  F.     Minute  par- 
ticles of  air  are  necessary  to  start  evaporation  sooner;  their  function  is  probably 
to  aid  in  the  diffusion  of  heat. 

(1)  Tyndall :  Heat  as  a  Mode  of  Motion.  (2)  Nichols  and  Franklin  :  The  Ele- 
ments of  Physics,  I,  161.  (3)  See  paper  by  J.  E.  Siebel :  The  Molecular  Constitu- 
tion of  Solids,  in  Science,  Nov.  5,  1909,  p.  654.  (4)  Quarterly  Mathematical  Journal 
April,  1855.  (5)  Darling:  Heat  for  Engineers,  208. 


10  APPLIED  THERMODYNAMICS 


SYNOPSIS   OF   CHAPTER  I 

Heat  is  the  universal  source  of  motive  power. 

Theories  of  heat :  the  caloric  theory  —  heat  is  matter ;  the  mechanical  theory  —  heat 

is  molecular  motion,  mutually  convertible  with  mechanical  energy. 
THERMOCHEMISTRY,  THERMODYNAMICS. 
Thermodynamics :  the  mechanical  theory  of  heat ;  in  its  engineering  applications,  the 

science  of  heat-motor  efficiency. 

Heat  intensity,  temperature  :  definition  of,  measurement  of  ;  pyrometers. 
Thermometric  scales :    Fahrenheit,    Centigrade,    Reaumur ;    fixed   points   and   their 

variations. 

Cardinal  properties :  pressure,  volume,  temperature  ;  PV=  (/)  T. 
Assumptions:  uniform  thermal  condition ;  no  chemical  action  ;  mechanical  equilibrium. 
Effects  of  heat :  H=T+  1+  W;  T+  I="  internal  energy  "  ;   W  =  external  work. 
Changes  of  physical  state,  perceptible  and  imperceptible  :  I  =  disgregation  work. 
Solid,   liquid,   vapor,   gas :    melting    point,    boiling   point ;    insensible   evaporation ; 

sublimation. 

PROBLEMS 

1.  Compute  the  freezing  points,  on  the  Centigrade  scale,  of  mercury  and  alcohol. 

2.  At  what  temperatures,  Reaumur,  do  alcohol  and  mercury  boil  ? 

3.  The  normal  temperature  of  the  human  body  is  98.6°  F.     Express  in  Centigrade 


4.  At,  what  temperatures  do  the  Fahrenheit  and  Centigrade  thermometers  read 
alike  ? 

5.  At  what  temperatures  do  the  Fahrenheit  and  Reaumur  thermometers  read 
alike  ? 

6.  Express  the  temperature  —  273°  C.  t>n  the  Fahrenheit  and  Reaumur  scales. 


CHAPTER  II 

THE   HEAT   UNIT:     SPECIFIC   HEAT:     FIRST   LAW  OF 
THERMODYNAMICS 

19.  Temperature  —  Waterfall  Analogy.     The  difference  between  temperature 
and  quantity  of  heat  may  be  apprehended  from  the  analogy  of  a  waterfall.     Tem- 
perature is  like  the  head  of  water ;  the  energy  of  the  fall  depends  upon  the  head, 
but  cannot  be  computed  without  knowing  at  the  same  time  the  quantity  of  water. 
As  waterfalls  of  equal  height  may  differ  in  power,  while  those  of  equal  power  may 
differ  in  fall,  so  bodies  at  like  temperatures  may  contain  different  quantities  of 
heat,  and  those  at  unequal  temperatures  may  be  equal  in  heat  contents. 

20.  Temperatures  and  Heat  Quantities.     If  we   mix  equal  weights  of 
water  at  different  temperatures,  the  resulting  temperature  of  the    mix- 
ture will  be  very  nearly  a  mean  between  the  two  initial  temperatures. 
If  the  original  weights  are  unequal,  then  the  final  temperature  will  be 
nearer  that  initially  held  by  the  greater  weight.     The  general  principle  of 
transfer  is  that 

The  loss  of  heat  by  the  hotter  water  will  equal  the  gain  of  heat  by  the 
colder. 

Thus,  5  Ib.  of  water  at  200°  mixed  with  1  Ib.  at  104°  gives  6  Ib.  at 
184°;  the  hotter  water  having  lost  80  "pound-degrees,"  and  the  colder 
water  having  gained  the  same  amount  of  heat.  If,  however,  we  mix  the 
5  Ib.  of  hot  water  with  1  Ib.  of  some  other  substance  —  say  linseed  oil  — 
the  resulting  temperature  will  not  be  184°,  but  194?6°,  if  the  initial  tem- 
perature of  the  oil  is  104°. 

21.  General  Principles.     Before  proceeding,  we  may  note,  in  addition  to  the 
principle  just  laid  down,  the  following  laws  which  are  made  apparent  by  the  ex- 
periments described  and  others  of  a  similar  nature  : 

(a)  In  a  homogeneous  substance,  the  movement  of  heat  accom- 
panying a  given   change   of   temperature  *  is    proportional  to  the 
weight  of  the  substance. 

(b)  The  movement  of  heat  corresponding  to  a  given  change  of 

*  Not  only  the  amount,  but  the  method,  of  changing  the  temperature  must  be 
fixed  (Art.  57). 

31 


12  APPLIED  THERMODYNAMICS 

temperature  is  not  necessarily  the  same  for  equal  intervals  at  all 
parts  of  the  thermometric  scale ;  thus,  water  cooling  from  200°  to 
195°  does  not  give  out  exactly  the  same  quantity  of  heat  as  in  cool- 
ing from  100°  to  95°. 

(<?)  The  loss  of  heat  during  cooling  through  a  stated  range  of 
temperature  is  exactly  equal  to  the  gain  of  heat  during  warming 
through  the  same  range. 

22.  The  Heat  Unit.     Changes  of  temperature  alone  do  not  measure  heat  quan- 
tities, because  heat  produces  other  effects  than  that  of  temperature  change.     If, 
however,  we  place  a  body  under  "standard"  conditions,  at  which  these  other 
effects,  if  not  known,  are  at  least  constant,  then  we  may  define  a  unit  of  quantity 
of  heat  by  reference  to  the  change  in  temperature  which  it  produces,  understand- 
ing that  there  may  be  included  perceptible  or  imperceptible  changes  of  other 
kinds,  not  affecting  the  constancy  of  value  of  the  unit. 

The  British  thermal  Unit  is  that  quantity  of  heat  which  is  expended  in 
raising  the  temperature  of  one  pound  of  water  (or  in  producing  other  effects 
during  this  change  in  temperature)  from  62°  to  63°  F.* 

To  heat  water  over  this  range  of  temperature  requires  very  nearly  the  same 
expenditure  of  heat  as  is  necessary  to  warm  it  1°  at  any  point  on  the  thermometric 
scale.  In  fact,  some  writers  define  the  heat  unit  as  that  quantity  of  heat  necessary 
to  change  the  temperature  from  39.1°  (the  temperature  of  maximum  density)  to 
40.1°.  Others  use  the  ranges  32°  to  33°,  59°  to  60°,  or  39°  to  40°.  The  range  first 
given  is  that  most  recently  adopted. 

23.  French    Units.     The    French    or    C.  G.  S.    unit    of    heat    is    the 
calorie,  the  amount  of  heat  necessary  to  raise  the  temperature  of  one 
kilogram  of  water  1°  C.     Its  value  is  2.2046  x  f  =  3.96832  B.  t.  u.,  and 
1  B.  t.  u.  =  0.251996  cal.     The  calorie  is  variously  measured  from  4°  to 
5°  and  from  14.5°  to  15.5°  C.     The  gram-calorie  is  the  heat  required  to 
raise  the  temperature  of  one  gram  of  water  1°  C.     The  Centigrade  heat 
unit  measures  the  heat  necessary  to  raise  one  pound  of  water  1°  C.  in 
temperature. 

24.  Specific  Heat.     Reference  was  made  in  Art.  20  to  the  different  heat 
capacities  of  different  substances,  e.g.  water  and  linseed  oil.     If  we  mix 
a  stated  quantity  of  water  at  a  fixed  temperature  successively  with  equal 
weights  of  various  materials,  all  initially  at  the  same  temperature,  the 
final  temperatures  of  the  mixtures  will  all  differ,  indicating  that  a  unit 

*  There  are  certain  grounds  for  preferring  that  definition  which  makes  the  B.  t.  u. 
t!ie  T|-o  part  of  the  amount  of  heat  required  to  raise  the  temperature  of  one  pound  of 
water  at  atmospheric  pressure  from  the  freezing  point  to  the  boiling  point. 


THE  HEAT   UNIT.     SPECIFIC  HEAT  13 

rise  of  temperature  of  unit  weight  of  these  various  materials  represents  a 
different  expenditure,  of  heat  in  each  case. 

The  property  by  virtue  of  which  materials  differ  in  this  respect  is 
that  of  specific  heat,  which  may  be  defined  as  the  quantity  of  heat 
necessary  to  raise  the  temperature  of  unit  weight  of  a  body  through  one 
degree. 

The  specific  heat  of  water  at  standard  temperature  (Art.  22)  is,  meas- 
ured in  B.  t.  u.,  1.0 ;  generally  speaking,  its  value  is  slightly  variable,  as  is 
that  of  all  substances. 

Rankine's  definition  of  specific  heat  is  illustrative :  "  the  specific  heat  of  any 
substance  is  the  ratio  of  the  weight  of  water  at  or  near  39.1°  F.  [62°-63°  F.]  which 
has  its  temperature  altered  one  degree  by  the  transfer  of  a  given  quantity  of  heat, 
to  the  weight  of  the  other  substance  under  consideration,  which  has  its  temperature 
altered  one  degree  by  the  transfer  of  an  equal  quantity  of  heat." 

25.  Mixtures    of    Different    Bodies.     If    the    weights    of   a   group    of 
mixed  bodies  be  X,  T,  Z,  etc.,  their  specific  heats  x,  y,  z,  etc.,  their  ini- 
tial temperatures  t,  u,  v,  etc.,  and  the  final  temperature  of  the  mixture 
be  m,  then  we  have  the  following  as  a  general  equation  of  thermal  equi- 
librium, in  which  any  quantity  may  be  solved  for  as  an  unknown: 

xX(t  —  m)  +  y  T(u  —  m)  +  zZ(v  -  m)  •  •  •  =  0. 

This  illustrates  the  usual  method  of  ascertaining  the  specific  heat  of  any 
body.  When  all  the  specific  heats  are  known,  the  loss  of  heat  to  sur- 
rounding bodies  may  be  ascertained  by  introducing  the  additional  term, 
+  R,  on  the  left-hand  side  of  this  equation.  The  solution  will  usually 
give  a  negative  value  for  R,  indicating  that  surrounding  bodies  have 
absorbed  rather  than  contributed  heat. .  The  value  of  R  will  of  course  be 
expressed  in  heat  units. 

26.  Specific   Heat  of   Water.     The   specific  heat  of   water,  according 
to   Rowland's   experiments;   decreases  as   the   temperature   is   increased 
from  39.1°  to  80°  F.,  at  which  latter  temperature  it  reaches  a  minimum 
value,  afterward  increasing   (Art.  359,  footnote).     The   variation  in  its 
value  is  very  small.     The  approximate  specific  heat,  1.0,  is  high  as  com- 
pared with  that  of  almost  all  other  substances. 

27.  Problems  Involving  Specific  Heat.     The  quantity  of  heat  re- 
quired to  produce  a  given  change  of  temperature  in  a  body  is  equal 
to  the  weight  of  the  body,  multiplied  by  the  range  of  temperature 
and  by  the  specific  heat. 

Or,  symbolically,  using  the  notation  of  Art.  25, 

H=xX(m-t). 


14  APPLIED  THERMODYNAMICS 

If  the  body  is  cooled,  then  m,  the  final  temperature,  is  less  than  £,  and  the  sign  of 
H  is  —  ;  if  the  body  is  wanned,  the  sign  of  //is  +  ,  indicating  a  reception  of  heat. 

28.  Consequences  of  the  Mechanical  Theory.    The  Mechanical  Equivalent 
of   Heat.      Even    before    Joule's    formulation   (Art.    2),    Rumford's    ex- 
periments  had   sufficed   for   a   comparison    of    certain    effects    of    heat 
with  an  expenditure  of  mechanical  energy.      The  power  exerted  by  the 
Bavarian  horses  used  to  drive  his  machinery  is  uncertain ;  but  Alexander 
has  computed  the  approximate  relation  to  have  been  847  foot-pounds  = 
1>B.  t.  u.  (1),  while  another  writer  fixes  the  ratio  at  1034,  and  Joule  cal- 
culated the  value  obtained  to  have  been  849. 

Carnot's  work,  although  based  throughout  on  the  caloric  theory,  shows  evident 
doubts  as  to  its  validity.  This  writer  suggested  (1824)  a  repetition  of  Kurnford's 
experiments,  with  provision  for  accurately  measuring  the  force  employed.  Using 
a  method  later  employed  by  Mayer  (Art.  29)  he  calculated  that  "0.611  units 
of  motive  power "  were  equivalent  to  "  550  units  of  heat " ;  a  relation  which 
Tyndall  computes  as  representing  370  kilogram-meters  per  calorie,  or  676  foot- 
pounds per  B.  t.  u.  Montgolfier  and  Seguin  (1839)  may  possibly  have  anticipated 
Mayer's  analysis. 

29.  Mayer's  Calculation.     This  obscure  German  physician  published  in  1842 
(2)  his  calculation  of  the  mechanical  equivalent  of  heat,  based  on  the  difference 
in  the  specific  heats  of  air  at  constant  pressure  and  constant  volume,  giving 
the  ratio  771.4  foot-pounds  per  B.  t.  u.  (Art.  72).     This  was  a  substantially  correct 
result,  though  given  little  consideration  at  the  time.     Mayer  had  previously  made 
rough  calculations  of  equivalence,  one  being  based  on  the  rise  of  temperature 
occurring  in  the  "  beaters  "  of  a  paper  mill. 

30.  Joule's  Determination.  Joule,  in  1843,  presented  the  first  of  his 
exhaustive  papers  on  the  subject.  The  usual  form  of  apparatus  employed 
has  been  shown  in  Fig.  1.  In  the  appendix  to  his  paper  Joule  gave  770  as 
the  best  value  deducible  from  his  experiments.  In  1849  (3)  he  presented 
the  figure  for  many  years  afterward  accepted  as  final,  viz.  772. 

In  1878  an  entirely  new  set  of  experiments  led  to  the  value  772.55,  which 
Joule  regarded  as  probably  slightly  too  low.  Experiments  in  1857  had  given  the 
values  745,  753,  and  766.  Most  of  the  tests  were  made  with  water  at  about  60°  F. 
This,  with  the  value  of  g  at  Manchester,  where  the  experiments  were  made,  in- 
volves slight  corrections  to  reduce  the  results  to  standard  conditions  (4). 

31.  Other  Investigators.  Of  independent,  though  uncertain,  merit,  were  the 
results  deduced  by  the  Danish  engineer,  Colding,  in  1843.  His  value  of  the 
equivalent  is  given  by  Tyndall  as  638  (5).  Helmholtz  (1847)  treated  the  matter 
of  equivalence  from  a  speculative  standpoint.  Assuming  that  "perpetual  motion  " 
is  impossible,  he  contended  that  there  must  be  a  definite  relation  between  heat 
energy  and  mechanical  energy.  As  early  as  1845,  Holtzmann  (6)  had  apparently 


MECHANICAL  EQUIVALENT   OF  HEAT  15 

independently  calculated  the  equivalence  by  Mayer's  method.  By  1847  the  reality 
of  the  numerical  relation  had  been  so  thoroughly  established  that  little  more  was 
heard  of  the  caloric  theory.  Clausius,  following  Mayer,  in  1850  obtained  wide 
circulation  for  the  value- 758  (7). 

32.  Hirn's  Investigation.     Joule  had  employed  mechanical  agencies  in  the 
heating  of  water.     Him,  in  1865  (8),  described  an  experiment  by  which  he  trans- 
formed into  heat  the  work   expended  in  producing  the  impact  of  solid  bodies. 
Two  blocks,  one  of  iron,  the  other  of  wood,  faced  with  iron  in  contact  with  a  lead 
cylinder,  were  suspended  side  by  side  as  pendulums.     The  iron  block  was  allowed 
to  strike  against  the  wood  block  and  the  rise  in  temperature  of  water  contained  in 
the  lead  cylinder  was  noted  and  compared  with  the  computed  energy  of  impact. 
The  value  obtained  for  the  equivalent  was  775. 

Far  more  conclusive,  though  less  accurate,  results  were  obtained 
by  Him  by  noting  that  the  heat  in  the  exhaust  steam  from  an  engine 
cylinder  was  less  than  that  which  was  present  in  the  entering  steam. 
It  was  shown  by  Clausius  that  the  heat  which  had  disappeared  was 
always  roughly  proportional  to  the  work  done  by  the  engine,  the 
average  ratio  of  foot-pounds  to  heat  units  being  753  to  1.  This  was 
virtually  a  reversal  of  Joule's  experiment,  illustrating  as  it  did  the 
conversion  of  heat  into  work.  It  is  the  most  striking  proof  we  have 
of  the  equivalence  of  work  and  heat. 

33.  Recent  Practice.     In  1876  a  committee  of  the  British  Association  for  the 
Advancement  of  Science  reviewed  critically  the  work  of  Joule,  and  as  a  mean 
value,  derived  from  his  best  60  experiments,  recommended  the  use  of  the  figure 
774.1,  which  was  computed  to  be  correct  within  ?fa.     In  1879,  Rowland,  having 
conducted  exact  experiments  on  the  specific  heat  of  water,  carefully  redetermiued 
the  value  of  the  equivalent  by  driving  a  paddle  wheel  about  a  vertical  axis  at 
fixed  speed,  in  a  vessel  of  water  prevented  from  turning  by  counterbalance  weights. 
The  torque  exerted  by  the  paddle  was  measured.     This  permitted  of  a  calculation 
of  the  energy  expended,  which  was  compared  with  the  rise  in  temperature  of  the 
water.     Rowland's  value  was  778,  with  water  at  its  maximum  density.      This 
was  regarded  as  possibly  slightly  low  (9).     Since  the  date  of  Rowland's  work,  the 
subject  has  been  investigated  by  Griffiths  (10),  who  makes  the  value  somewhat 
greater  than  778,  and  by  Reynolds  and  Moorby  (11),  who  report  the  ratio  778  as 
the  mean  obtained  for  a  range  of  temperature  from  32°  to  212°  F.     This  they 
regard  as  possibly  1  or  2  foot-pounds  too  low. 

34.  Summary.     The  establishing  of  a  definite  mechanical  equivalent  of 
heat  may  be  regarded  as  the  foundation  stone  of  thermodynamics.    Accord- 
ing to  Merz  (12),  the  anticipation  of  such  an  equivalent  is  due  to  Poucelet 
and  Carnot ;  Rumford's  name  might  be  added.     "  The  first  philosophical 
generalizations  were  given  by  Mohr  and  Mayer;  the  first  mathematical 


16  APPLIED  THERMODYNAMICS 

treatment  by  Helmholtz;  the  first  satisfactory  experimental  verification 
by  Joule.7'  The  construction  of  the  modern  science  on  this  foundation 
has  been  the  work  chiefly  of  Rankine,  Clausius,  and  Kelvin. 

35.  First  Law  of  Thermodynamics.     Heat  and  mechanical  energy 
are  mutually  convertible  in  the  ratio  of  778  foot-pounds  to  the  British 
thermal  unit. 

This  is  a  restricted  statement  of  the  general  principle  of  the  conservation  of 
energy,  a  principle  which  is  itself  probably  not  susceptible  to  proof. 
We  have  four  distinct  proofs  of  the  first  law  : 

(a)  Joule's  and  Rowland's  experiments  on  the  production  of 
heat  by  mechanical  work. 

(5)  Hirn's  observations  on  the  production  of  work  by  the  ex- 
penditure of  heat. 

(0)  The  computations  of  Mayer  and  others,  from  general  data. 

(d)  The  fact  that  the  law  enables  us  to  predict  thermal  proper- 
ties of  substances  which  experiments  confirm. 

36.  Wormell's  Theorem.     There   cannot  'be   two  values  of  the   mechanical 
equivalent  of  heat.     Consider  two  machines,  si  and  B,  in  the  first  of  which  work 
is  transformed  into  heat,  and  in  the  second  of  which  heat  is  transformed  into 
work.     Let  /  be  the  mechanical  equivalent  of  heat  for  A,  W  the  amount  of  work 
which  it  consumes  in  producing  the  heat  Q;  then  W  =  JQ  or  Q  =  W  -4-  J.     Let 
this  heat  Q  be  used  to  drive  the  machine  B,  in  which  the  mechanical  equivalent 
of  heat  is,  say  K.     Then  the  work  done  by  B  is  V  =  KQ  =  KW  -=-  J.     Let  this 
work  be  now  expended  in  driving  A.     It  will  produce  heat  R,  such  that  JR  =  V 
or  R  =  V  -f-  J.     If  this  heat  R  be  used  in  B,  work  will  be  done  equal  to  KR  ;  but 

KR  =  KV  +  J  =  (—\*  W. 


Similarly,  after  n  complete  periods  of  operation,  all  parts  of  the  machines  occupy- 
ing the  same  positions  as  at  the  beginning,  the  work  ultimately  done  by  B  will  be 

^Yw. 


If  K  is  less  than  /,  this  expression  will  decrease  as  n  increases;  i.e.  the  system 
will  tend  continually  to  a  state  of  rest,  contrary  to  the  first  law  of  motion.  If  K 
be  greater  than  J,  then  as  n  increases  the  work  constantly  increases,  involving  the 
assumed  fallacy  of  perpetual  motion.  Hence  K  and  J  must  be  equal  (13). 

37.  Significance  of  the  Mechanical  Equivalent.  A  very  little  heat  is  seen  to  be 
equivalent  to  a  great  deal  of  work.  The  heat  used  in  raising  the  temperature  of 
one  pound  of  water  100°  represents  energy  sufficient  to'lift  one  ton  of  water  nearly 
39  feet.  The  heat  employed  to  boil  one  pound  of  water  initially  at  32°  F.  would 


FIRST  LAW  OF  THERMODYNAMICS  17 

suffice  to  lift  one  ton  443  feet.    The  heat  evolved  in  the  combustion  of  one  pound  of 
hydrogen  (62,000  B.  t.  u.)  would  lift  one  ton  nearly  five  miles. 

(1)  Treatise  on  Thermodynamics,  London,  1892.  (2)  Wohler  and  Liebig's 
Annalen  der  Pharmacie :  Bemerkungen  uber  die  Krafte  der  unbelebten  Natur,  May, 
1842.  (3)  Phil.  Trans.,  1850.  (4)  Joule's  Scientific  Papers,  Physical  Society  of 
London,  1884.  (5)  Probably  quoted  by  Tyndall  from  a  later  article  by  Colding,  in 
which  this  figure  is  given.  Colding's  original  paper  does  not  seem  to  be  accessible. 

(6)  Ueber  die   Wdrme    und  Elasticitdt   der   Gase    und  Dampfe,   Mannheim,    1845. 

(7)  Poggendorff.  Annalen,  1850.     (8)  Theorie  Mecanique,  etc.,  Paris,  1865.     (9)  Proc. 
Amer.  Acad.  Arts  and  Sciences,  New  Series,  VII,  1878-79.     (10)  Phil.  Trans.  Boy. 
Soc.,  1893.     (11)  Phil.   Trans.,  1897.     (12)  History  of  European   Thought,  II,  137. 
(13)  R.  Wormell:  Thermodynamics,  1886. 


SYNOPSIS  OF  CHAPTER  H 

Heat  and  temperature  :  heat  quantity  vs.  heat  intensity. 

Principles :  (a)  heat  movement  proportional  to  weight  of  substance  ;  (&)  temperature 
range  does  not  accurately  measure  heat  movement ;  (c)  loss  during  cooling  equals 
gain  during  warming,  for  identical  ranges. 

The  British  thermal  unit:  other  units  of  heat  quantity. 

Specific  heat :  mixtures  of  bodies  ;  quantity  of  heat  to  produce  a  given  change  of  tem- 
perature ;  specific  heat  of  water. 

The  mechanical  equivalent  of  heat :  early  approximations.  First  law  of  thermody- 
namics :  proofs ;  only  one  value  possible  ;  examples  of  the  motive  power  of  heat. 


PROBLEMS 

1.  How  many  Centigrade  heat  units  are  equivalent  to  one  calorie  ? 

2.  Find  the  number  of  gram-calories  in  one  B.t.  u. 

3.  A  mixture  is  made  of  5  Ib.  of  water  at  200°,  3  Ib.  of  linseed  oil  at  110°,  and 
22  Ib.  of  iron  at  220°,  the  respective  specific  heats  being  1.0,  0.3,  and  0.12.    Find  the 
final  temperature,  if  no  loss  occurs  by  radiation. 

4.  If  the  final  temperature  of  the  mixture  in  Problem  3  is  189°  F.,  find  the  num- 
ber of  heat  units  lost  by  radiation. 

5.  Under  what  conditions,  in  Problem  3,  might  the  final  temperature  exceed  that 
computed  ? 

6.  How  much  heat  is  given  out  by  7£  Ib.  of  linseed  oil  in  cooling  from  400°  F.  to 
32°  F.  ? 

7.  In  a  heat  engine  test,  each  pound  of  steam  leaves  the  engine  containing  125.2 
B.t.u.  less  heat  than  when  it  entered  the  cylinder.     The  engine  develops  155  horse- 
power, and  consumes  3160  Ib.  of  steam  per  hour.     Compute  the  mechanical  equivalent 
of  heat. 

8.  A  pound  of  good  coal  will  evolve  14,000  B.t.u.     Assuming  a  train  resistance 
of  11  Ib.  per  ton  of  train  load,  how  far  should  one  ton  (2000  Ib.)  of  coal,  burned  in  the 
locomotive  without  loss,  propel  a  train  weighing  2000  tons  ?    If  the  locomotive  weighs 
125  tons,  how  high  would  one  pound  of  coal  lift  it,  if  fully  utilized? 


18  APPLIED  THERMODYNAMICS 

9.   Find  the  number  of  kilogram-meters  equivalent  to  1  calorie.     (1  meter  =  39.37 
in.,  1  kilogram  =  2.2046  Ib.) 

10.  Transform  the  following  formula  (P  being  the  pressure  in  kilograms  per  square 
meter,  Fthe  volume  in  cubic  meters  per  kilogram,  !Tthe  Centigrade  temperature  plus 
273),  to  English  units,  letting  the  pressure  be  in  pounds  per  square  inch,  the  volume 
in  cubic  feet  per  pound,  and  the  temperature  that  on  the  Fahrenheit  scale  plus  459.4, 
and  eliminating  coefficients  in  places  where  they  do  not  appear  in  the  original  equation  : 


PV=  47.1  T-  P(l  +  0.000002  PO.OSl  -  0.0052 


CHAPTER  III 

LAWS  OF  GASES :  ABSOLUTE  TEMPERATURE :  THE  PERFECT  GAS 

38.  Boyle's  (or  Mariotte's)   Law.     The  simplest   thermodynamic 
relations  are  those  exemplified  by  the  so-called   permanent   gases. 
Boyle  (Oxford,  1662)  and  Mariotte  (1676-1679)  separately  enun- 
ciated the  principle  that  at  constant  temperature  the  volumes  of  gases 
are  inversely  proportional  to  their  pressures.     In  other  words,  the 
product  of  the  specific  volume  and  the  pressure  of  a  gas  at  a  given 
temperature  is  a  constant.     For  air,  which  at  32°  F.  has  a  volume 
of  12.387  cubic  feet  per  pound  when  at  normal  atmospheric  pressure, 
the  value  of  the  constant  is,/0r  this  temperature, 

144  x  14.7  x  12.387  =  26,221. 

Symbolically,  if  c  denotes  the  constant  for  any  given  tempera- 
ture, 

pv  =  P  V  or,  pv  =  c. 

Figure  2  represents  Boyle's  law  graphically,  the  ordinates  being  pres- 
sures per  square  foot,  and  the  abscissas,  volumes  in  cubic  feet  per  pound. 
The  curves  are  a  series  of  equilateral  hyperbolas,  plotted  from  the  second 
of  the  equations  just  given,  with  various  values  of  c. 

39.  Deviations  from  Boyle's  Law.     This  experimentally  determined  principle 
was  at  first  thought  to  apply  rigorously  to  all  true  gases.     It  is  now  known  to  be 
not  strictly  correct  for  any  of  them,  although  very  nearly  so  for  air,  hydrogen, 
nitrogen,  oxygen,  and  some  others.     All  gases  may  be  liquefied,  and  all  liquids 
may  be  gasified.     When  far  from  the  point  of  liquefaction,  gases  follow  Boyle's 
law.     When  brought  near  the  liquefying  point  by  the  combined  influences  of  high 
pressure  and  low  temperature,  they  depart  widely  from  it.     The  four  gases  just 
mentioned  ordinarily  occur  at  far  higher  temperatures  than  those  at  which  they 
will  liquefy.     Steam,  carbon  dioxide,  ammonia  vapor,  and  some  other  well-known 
gaseous   substances  which   may  easily  be  liquefied   do   not  follow  the  law  even 
approximately.     Conformity  with  Boyle's  law  may  be  regarded  as  a  measure  of 
the  "  perfectness  "  of  a  gas,  or  of  its  approximation  to  the  truly  gaseous  condition. 

19 


20 


APPLIED  THERMODYNAMICS 


2WO 


10  20  30  40  50 

FIG.  2.     Arts.  38,  91.  —  Boyle's  Law. 

40.  Dalton's  Law,  Avogadro's  Principle.  Dalton  has  been  credited  (though 
erroneously)  with  the  announcement  of  the  law  now  known  as  that  of  Gay-Lussac 
or  Charles  (Art.  41).  What  is  properly  known  as  Dalton's  law  may  be  thus 
stated :  A  mixture  of  gases  having  no  chemical  action  on  one  another  exerts  a  pres- 
sure which  is  the  sum  of  the  pressures  which  would  be  exerted  by  the  component 
gases  separately  if  each  in  turn  occupied  the  containing  vessel  alone  at  the  given 
temperature. 

The  ratio  of  volumes,  at  standard  temperature  and  pressure,  in  which  two 
gases  combine  chemically  is  always  a  simple  rational  fraction  ($,  f,  f,  etc.). 
Taken  in  conjunction  with  the  molecular  theory  of  chemical  combination,  this 
law  leads  to  the  principle  of  Avogadro  that  all  gases  contain  the  same  number  of 
molecules  per  unit  of  volume,  at  the  same  temperature  and  pressure.  This  law  has 
important  thermodynamic  relations. 


41.  Law  of  Gay-Lussac  or  of  Charles  (1).  Davy  had  announced  that  the 
coefficient  of  expansion  of  air  was  independent  of  the  pressure.  Gay-Lus- 
sac verified  this  by  the  apparatus  shown  in  Fig.  3.  He  employed  a  glass 
tube  with  a  large  reservoir  A,  containing  the  air,  which  had  been  previously 


LAWS  OF   GASES 


21 


dried.  An  index  of  mercury  mn  separated  the  air  from  the  external  atmos- 
phere, while  permitting  it  to  expand.  The  vessel  B  was  first  filled  with 
melting  ice.  Upon  applying  heat,  equal  in- 
tervals of  temperature  shown  on  the  ther- 
mometer C  were  found  to  correspond  with 
equal  displacements  of  the  index  mn.  When 
a  pressure  was  applied  on  the  atmospheric 
side  of  the  index,  the  proportionate  expansion 
of  the  air  was  shown  to  be  still  constant  for 
equal  intervals  of  temperature,  and  to  be  equal 
to  that  observed  under  atmospheric  pressure. 
Precisely  the  same  results  were  obtained  with  FIG.  3.  Arts.  41,48.— Verifica- 
other  gases.  The  expansion  of  dry  air  was 

found  to  be  0.00375,  or  -^iy  °^  ^ne  v°lume  at  the  freezing  point,  for  each 
degree  C.  of  rise  of  temperature.  The  law  thus  established  may  be 
expressed : 

For  all  gases,  and  at  any  pressure,  maintained  constant,  equal  increments  of 
volume  accompany  equal  increments  of  temperature. 

42.    Increase  of  Pressure  at  Constant  Volume.     A  second  statement 
of  this  law  is  that  all  gases,  when  maintained  at  constant  volume, 

undergo  equal  increases  of 
pressure  with  equal  increases 
of  temperature. 


Bfl 


FIG.  4.    Arts.  42,  48.  —  Coefficient  of  Pressure. 


This  is  shown  experimen- 
tally by  the  apparatus  of  Fig.  4. 
The  glass  bulb  A  contains  the 
gas.  It  communicates  with  the 
open  tube  manometer  Mm, 
which  measures  the  pressure 
P  is  a  tube  containing  mercury, 

in  which  an  iron  rod  is  submerged  to  a  sufficient  depth  to  keep  the  level 
of  the  mercury  in  m  at  the  marked  point  a,  thus  maintaining  a  constant 
volume  of  gas. 

43.  Regnault's  Experiments.  The  constant  0.00375  obtained  by  Gay- 
Lussac  was  pointed  out  by  Rudberg  to  be  probably  slightly  inaccurate. 
Regnault,  by  employing  four  distinct  methods,  one  of  which  was  sub- 
stantially that  just  described,  determined  accurately  the  coefficient  of 
increase  of  pressure,  and  finally  the  coefficient  of  expansion  at  constant 
pressure,  which  for  dry  air  was  found  to  be  0.003665,  or  Yfj,  per  degree 
C.,  of  the  volume  at  the  freezing  point. 


22  APPLIED  THERMODYNAMICS 

44.  Graphical  Representation.     In  Fig.  5,  let  ab  represent  the 
volume  of  a  pound  of  gas  at  32°  F.     Let  temperatures  and  volumes 

be  represented,  respectively,  by  ordinates  and 
abscissas.  According  to  Charles'  Law,  if  the 
pressure  be  constant,  the  volumes  and  tempera- 
tures will  increase  proportionately  ;  the  volume 
ab  increasing  ^yg-  for  each  degree  C.  that  the 
temperature  is  increased,  and  vice  versa.  The 
straight  line  cbe  then  represents  the  successive 
relations  of  volume  and  temperature  as  the  gas 

FIG.  5.  Arts.  44,  84.—  is  heated  or  cooled  from  the  temperature  at  b. 
Charles'  Law.  ^  j.|ie  pOjnt  c,  where  this  line  meets  the  a  T  axis, 

the  volume  of  the  gas  will  be  zero,  and  its  temperature  will  be  273°  C., 

or  491.4°  F.,  below  the  freezing  point. 

45.  Absolute  Zero.     This  temperature  of     -  459.4°  F.  suggests 
the  absolute  zero  of  thermodynamics.     All  gases  would  liquefy  or 
even  solidify  before  reaching  it.     The  lowest  temperature   as  yet 
attained  is  about  450°  F.  below  zero.     The  absolute  zero  thus  experi- 
mentally conceived  (a  more  strictly  absolute  scale  is  discussed  later, 
Art.  156)  furnishes   a   convenient  starting  point  for  the  measure- 
ment of  temperature,  which  will  be  employed,  unless  otherwise  speci- 
fied, in  our  remaining  discussion.     Absolute   temperatures  are  those 
in  which  the  zero  point  is  the  absolute  zero.     Their  numerical  values 
are  to  be  taken,  for  the  present,  at  459.4°  greater  than  those  of  the  cor- 
responding Fahrenheit  temperature. 

46.  Symbolical  Representation.     The  coefficients  determined  by  Gay-Lussac, 
Charles,  and  Regnault  were  those  for  expansion  from  an  initial  volume  of  32°  F. 
If  we  take  the  volume  at  this  temperature  as  unity,  then  letting  T  represent  the 
absolute  temperature,  we  have,  for  the  volume  at  any  temperature, 

V=  r-491.4. 

Similarly,  for  the  variation  in  pressure  at  constant  volume,  the  initial  pressure 
being  unity,  P  =  T  +  491.4.  If  we  let  a  denote  the  value  1  -491.4,  the  first 
expression  becomes  V  -  aT,  and  the  second,  P  =  aT.  Denoting  temperatures  on 
the  Fahrenheit  scale  by  t,  we  obtain,  for  an  initial  volume  v  at  32°  and  any  other 
volume  F  corresponding  to  the  temperature  t,  produced  without  change  of  pressure, 

F=»[l +  a(*- 32)]. 

Similarly,  for  variations  in  pressure  at  constant  volume, 
P  =/>[!+•«(' -32)]. 


LAWS   OF   GASES:    ABSOLUTE  TEMPERATURE         23 

The  value  of  a  is  experimentally  determined  to  be  very  nearly  the  same  for  pres- 
sure changes  as  for  volume  changes  ;  the  difference  in  the  case  of  air  being  less 
than  \  of  one  per  cent.  The  temperature  interval  between  the  melting  of  ice  and 
the  boiling  of  water  being  180°,  the  expansion  of  volume  of  a  gas  between  those 

=  0.365,  whence  Rankine's  equation,  originally  derived  from  the 


limits  is 


18D  x  1 


491.4 
experiments  of  Regnault  and  Rudberg, 

PV 

pc 


=  1.365, 


in  which  7J,  V  refer  to  the  higher  temperature,  and  p,  v  to  the  lower. 


47.  Deviations  from  Charles'  Law.  The  laws  thus  enunciated  are  now  known 
not  to  hold  rigidly  for  any  actual  gases.  For  hydrogen,  nitrogen,  oxygen,  air, 
carbon  monoxide,  methane,  nitric  oxide,  and  a  few  others,  the  disagreement  is 
ordinarily  very  slight.  For  carbon  dioxide,  steam,  and  ammonia,  it  is  quite  pro- 
nounced. The  reason  for  this  is  that  stated  in  Art.  39.  The  first  four  gases  named 
have  expansive  coefficients,  not  only  almost  unvarying,  but  almost  exactly  identical. 
They  may  be  regarded  as  our  most  nearly  perfect  gases.  For  air,  for  example, 
Regnault  found  over  a  range  of  temperature  of  180°  F.,  and  a  range  of  pressure 
of  from  109.72  mm.  to  4992.09  mm.,  an  extreme  variation  in  the 
coefficients  of  only  1.67  per  cent.  For  carbon  dioxide,  on  the 
other  hand,  with  the  same  range  of  temperatures  and  a  de- 
creased pressure  range 
of  from  785.47  mm. 
to  4759.03  mm.,  the 
variation  was  4.72 
per  cent  of  the  lower 
value  (2). 


\ 


48.  The  Air  Thermometer.  The  law  of  Charles  sug- 
gests a  form  of  thermometer  far  more  accurate  than  the 
ordinary  mercurial  instrument. 
If  we  allow  air  to  expand  with- 
out change  in  pressure,  or  to 
increase  its  pressure  without 
change  in  volume,  then  we  have 
by  measurement  of  the  volume 
or  of  the  pressure  respectively  a 
direct  indication  of  absolute  tem- 
perature. The  apparatus  used 
by  Gay-Lussac  (Fig.  3),  or, 

equally,  that   shown  in  Fig.  4,  is  in  fact  an  air  ther- 
mometer, requiring  only  the  establishment  of  a  scale  to  fit  it  for  practical 
use.     A  simple  modern  form  of  air  thermometer  is  shown  in  Fig.  6.     The 


FIG.  6. 


Art.  48. —Air  Ther- 
mometer. 


FIG.  7.  Art.  48.  — 
Preston  Air 
Thermometer. 


24 


APPLIED  THERMODYNAMICS 


FIG.  8.  Art.  48. 
—  Hoadley 
Air  Ther- 
mometer. 


bulb  A  contains  dry  air,  and  communicates  through  a  tube 
of  fine  bore  with  the  short  arm  of  the  manometer  BB,  by 
means  of  which  the  pressure  is  measured.  The  level  of  the 
mercury  is  kept  constant  at  a  by  means  of  the  movable 
reservoir  R  and  flexible  tube  m.  The  Preston  air  ther- 
mometer is  shown  in  Fig.  7.  The  air  is  kept  at  constant 
volume  (at  the  mark  a)  by  admitting  mercury  from  the 
bottle  A  through  the  cock  B.  In  the  Hoadley  air  ther- 
mometer, Fig.  8,  no  attempt  is  made  to  keep  the  volume 
of  air  constant;  expansion  into  the  small  tube  below  the 
bulb  increasing  the  volume  so  slightly  that  the  error  is  com- 
puted not  to  exceed  5°  in  a  range  of  600°  (3). 

49.  Remarks  on   Air  Thermometers.      Following    Regnault, 
the  instrument  is   usually  constructed  to   measure  pressures   at 
constant  volume,   using  either  nitrogen,   hydrogen,   or  air  as   a 
medium.     Only  one  "  fixed  point "  need  be  marked,  that  of  the 
temperature  of  melting  ice.     Having  marked  at  32°  the  atmos- 
pheric pressure  registered  at  this  temperature,  the  degrees  are 
spaced  so  that  one  of  them  denotes  an  augmentation  of  pressure 
of  14.7  -r-  491.4  =  0.0299  Ib.  per  square  inch.     It  is  usually  more 
convenient,  however,  to  determine  the  two  fixed  points  as  usual 
and  subdivide  the  intervening   distance  into  180  equal  degrees. 
The  air  thermometer  readings  differ  to  some  extent  from  those  of 
the  most  accurate  mercurial   instruments,  principally  because  of 
the  fact  that  mercury  expands  much  less  than  any  gas,  and  the 
modifying  effect  of  the  expansion  of  the  glass  container  is  there- 
fore greater  in  its  case.     The   air  thermometer  is   itself  not  a 
perfectly  accurate  instrument,  since  air  does  not   exactly  follow 
Charles'  law  (Art.  47).     The  instrument  is  used  for  standardizing 
mercury  thermometers,  for  direct  measurements  of  temperatures 
below  the  melting  point  of  glass  (600-800°  F.),  as  in  Regnault's 
experiments  on  vapors;  or,  by  using  porcelain  bulbs,  for  measur- 
ing much  higher  temperatures. 

50.  The  Perfect  Gas.    If  actual  gases  conformed  pre- 
cisely to  the  laws  of  Boyle  and  Charles,  many  of  their 
thermal  'properties  might  be  computed  directly.     The 
slightness  of  the  deviations  which  actually  occur  sug- 
gests the  notion  of  a  perfect  gas,  which  would  exactly 
and  invariably  follow  the  laws, 

PV=c,   VP  =  aT,  PY  =  aT. 

Any  deductions  which  might  be  made  from  these  sym- 
bolical expressions  would  of  course  be  rigorously  true  only 


THE  PERFECT   GAS  25 

for  a  perfect  gas,  which  does  not  exist  in  nature.  The  current  tliermo- 
dynamic  method  is,  however,  to  investigate  the  properties  of  such  a  gas,  modi- 
fying the  results  obtained  so  as  to  make  them  applicable  to  actual  gases, 
rather  than  to  undertake  to  express  symbolically  or  graphically  as  a 
basis  for  computation  the  erratic  behavior  of  those  actual  gases.  The 
error  involved  in  assuming  air,  hydrogen,  and  other  "  permanent "  gases 
to  be  perfect  is  in  all  cases  too  small  to  be  of  importance  in  engineering 
applications.  Zeuner  (4)  has  developed  an  "  equation  of  condition "  or 
"characteristic  equation"  for  air  which  holds  even  for  those  extreme  con- 
ditions of  temperature  and  pressure  which  are  here  eliminated. 

51.  Properties  of  the  Perfect  Gas.     The  simplest  definition  is  that 
the  perfect  gas  is  one  which  exactly  follows  the  laws  of  Boyle  and 
Charles.     (Rankine's  definition  (5)  makes  conformity  to  Dalton's 
law  the  criterion  of  perfectness.)     Symbolically,  the  perfect  gas  con- 
forms to  the  law,  readily  deduced  from  Art.  50, 

PV=RT, 

in  which  R  is  a  constant  and  T  the  absolute  temperature.  Consid- 
ering air  as  perfect,  its  value  for  R  may  be  obtained  from  experi- 
mental data  at  atmospheric  pressure  and  freezing  temperature : 

R  =  PV+  ^=(14.7  x  144  x  12.387) -T- 491. 4  =  53.36  foot-pounds. 
For  other  gases  treated  as  perfect,  the  value  of  R  may  be  readily 
calculated  when  any  corresponding  specific  volumes,  pressures,  and 
temperatures  are  known.  Under  the  pressure  and  temperature  just 
assumed,  the  specific  volume  of  hydrogen  is  178.83  ;  of  nitrogen, 
12.75;  of  oxygeu,  11.20.  A  useful  form  of  the  perfect  gas  equation 
may  be  derived  from  that  just  given  by  noting  that  PV-*-  T=  R,  a 
constant :  PV  pv 

~T~~~~~  t  ' 

52.  Significance  of  R.     At  the  standard  pressure  and  temperature 
specified  in  Art.  51,  the  values  of   R  for  various   gases   are  obviously 
proportional  to  their  specific  volumes  or  inversely  proportional  to  their 
densities.     This  leads  to  the  form  of  the  characteristic  equation  some- 
times given,  PV  =  rT  -z-  M,  in  which  M  is  the  molecular  weight  and  r  a 
constant  having  the  same  value  for  all  sensibly  perfect  gases. 

53.  Molecular  Condition.     The  perfect  gas  is  one  in  which  the  molecules  move 
with  perfect  freedom,  the  distances  between  them  being  so  great  in  comparison 
with  their  diameters  that  no  mutually  attractive  forces  are  exerted.     No  per- 
formance of  disgregation  work  accompanies  changes  of  pressure  or  temperature. 


26  APPLIED  THERMODYNAMICS 

Hirschfeld  (6),  in  fact,  defines  the  perfect  gas  as  a  substance  existing  in  such  a 
physical  state  that  its  constituent  particles  exert  no  interattraction.  The  coefficient 
of  expansion,  according  to  Charles'  law,  would  be  the  exact  reciprocal  of  the  abso- 
lute temperature  of  melting  ice,  for  all  pressures  and  temperatures.  Zeuner  has 
shown  (7)  that  as  necessary  consequences  of  the  theory  of  perfect  gases  it  can  be 
proved  that  the  product  of  the  molecular  weight  and  specific  volume,  at  the  same 
pressure  and  temperature,  is  constant  for  all  gases;  whence  he  derives  Avogadro's 
principle  (Art.  40).  Rankine  (8)  has  tabulated  the  physical  properties  of  the 
"  perfect  gas." 

54.  Kinetic  Theory  of  Gases.     Beginning  with  Bernoulli!  in  1738,  various 
investigators  have  attempted  to  explain  the  phenomena  of  gases  on  the  basis  of 
the  kinetic  theory,  which  is  now  closely  allied  with  the  mechanical  theory  of  heat. 
According  to  the  former  theory,  the  molecules  of  any  gas  are  of  equal  mass  and 
like  each  other.     Those  of  different  gases  differ  in  proportions  or  structure.     The 
intervals  between  the  molecules  are  relatively  very  great.     Their  tendency  is  to 
move  with  uniform  velocity  in  straight  lines.     Upon  contact,  the  direction  of  mo- 
t.on  undergoes  a  change.     In  any  homogeneous  gas  or  mixture  of  gases,  the  mean 
energy  due  to  molecular  motion  is  the  same  at  all  parts.     The  pressure  of  the  gas 
per  unit  of  superficial  area  is  proportional  to  the  number  of  molecules  in  a  unit  of 
volume  and  to  the  average  energy  with  which  they  strike  this  area.     It  is  there- 
fore proportional  to  the  density  of  the  gas  and  to  the  average  of  the  squares  of  the 
molecular  velocities.     Temperature  is  proportional  to  the  average  kinetic  energy 
of  the  molecules.     The  more  nearly  perfect  the  gas,  the  more  infrequently  do  the 
molecules  collide  with  one  another.    When  a  containing  vessel  is  heated,  the  mole- 
cules rebound  with  increased  velocity,  and  the  temperature  of  the  gas  rises ;  when 
the  vessel  is  cooled,  the  molecular  velocity  and  the  temperature  are  decreased. 
"  When  a  gas  is  compressed  under  a  piston  in  a  cylinder,  the  particles  of  the  gas 
rebound  from  the  inwardly  moving  piston  with  unchanged  velocity  relative  to 
the  piston,  but  with  increased  actual  velocity,  and  the  temperature  of  the  gas  con- 
sequently rises.     When  a  gas  is  expanded  under  a  receding  piston,  the  particles  of 
the  gas  rebound  with  diminished  actual  velocity,  and  the  temperature  falls  "  (9). 

Recent  investigations  in  molecular  physics  have  led  to  a  new  terminology  but 
in  effect  serve  to  verify  and  explain  the  kinetic  theory. 

55.  Application  of  the  Kinetic  Theory.     Let  w  denote  the  actual  molecular 
velocity.    Resolve  this  into  components  x,  y,  and  z,  at  right  angles  to  one  another. 
Then  to2  -  x'2  +  y2  -f  z2.     Since  the  molecules  move  at  random  in  all  directions, 
x  =  y  =  z,  and  «o2  =  3x2.     Consider  a  single  molecule,  moving  in  an  x  direction 
back  and  forth  between  two  limiting  surfaces  distant  from  each  other  d,  the  x 
component  of  the  velocity  of  this  particle  being  a.     The   molecule  will  make 
(a  -+•  2  d)  oscillations  per  second.     At  each  impact  the  velocity  changes  from  +  a 
to  —  a,  or  by  2  a,  and  the   momentum  by  2  am,  if  m  represents  the  mass  of  the 
molecule.     The  average  rate  of  loss  of  momentum  is  2  am  x  (a  -h  2  d}  =  ma2  H-  d ; 
and  this  is  the  average  force  exerted  per  second  on  the  limiting  surfaces.     The 
total  force  exerted  by  all  of  the  molecules  on  these  surfaces  is  then  equal  to 

F  =  ^ N  =  ^  N  =  ^  N,  in  which  N  is  the  total  number  of  molecules  in  the 
d  d  '3d 


THE  PERFECT   GAS  27 

vessel.     Let  q  be  the  area  of  the  limiting  surface.     Then  the  force  per  unit  of  sur- 


face =  p=F+q=?gN+q=  ,  whence  pv  =  ;-^,  in  which  *  is  the  vol- 

o  a  o  v  3 

ume  of  the  gas  =  qd  (10). 

56.  Applications  to  Perfect  Gases.  Assuming  that  the  absolute  temperature 
is  proportional  to  the  average  kinetic  energy  per  molecule  (Art.  54),  this  kinetic 
energy  being  \  mw2-,  then  letting  the  mass  be  unity  and  denoting  by  R  a  constant 
relation,  we  have  pv  =  RT.  The  kinetic  theory  is  perfectly  consistent  with  Dal- 
tbn's  law  (Art.  40).  It  leads  also  to  Avogadro's  principle.  Let  two  gases  be  pres- 
ent. For  the  first  gas,  p  =  nmw2  -=-  3,  and  for  the  second,  P  =  NMW2  •*•  3.  If 
t  =  T,  mw2  =  M  IF2,  and  if  p  =  P,  then  n  =  N.  If  M  denote  the  mass  of  the  gas, 
M  —  mN,  and  pv  =  Mw2  -=-  3,  or  w2  =  3pv  -f-  M,  from  which  the  mean  velocity  of 
the  molecules  may  be  calculated  for  any  given  temperature. 

For  gases  not  perfect,  the  kinetic  theory  must  take  into  account,  (a)  the  effect 
of  occasional  collision  of  the  molecules,  and  (b)  the  effect  of  mutual  attractions 
and  repulsions.  The  effect  of  collisions  is  to  reduce  the  average  distance  moved 
between  impacts  and  to  increase  the  frequency  of  impact  and  consequently  the 
pressure.  The  result  is  much  as  if  the  volume  of  the  containing  vessel  were 
smaller  by  a  constant  amount,  b,  than  it  really  is.  For  v,  we  may  therefore  write 
v  —  b.  The  value  of  b  depends  upon  the  amount  and  nature  of  the  gas.  The 
effect  of  mutual  attractions  is  to  slow  down  the  molecules  as  they  approach  the 
walls.  This  makes  the  pressure  less  than  it  otherwise  would  be  by  an  amount 
which  can  be  shown  to  be  inversely  proportional  to  the  square  of  the  volume  of 
the  gas.  For  p,  we  therefore  write  p  +(«  -f-  v2),  in  which  a  depends  similarly 
upon  the  quantity  and  nature  of  the  gas.  We  have  then  the  equation  of  Van  der 
Waals, 

+(»  -*)=**'  (11)- 


(1)  Cf.  Verdet,  Lemons  de  Chemie  et  de  Physique,  Paris,  1862.  (2)  Rel.  des  Exp., 
I,  111,  112.  (3)  Trans.  A.  8.  M.  E.,  VI,  282.  (4)  Technical  Tliermodynamics 
(Klein  tr.),  II,  313.  (5)  "  A  perfect  gas  is  a  substance  in  such  a  condition  that  the 
total  pressure  exerted  by  any  number  of  portions  of  it,  against  the  sides  of  a  vessel  in 
which  they  are  inclosed,  is  the  sum  of  the  pressures  which  each  such  portion  would 
exert  if  enclosed  in  the  vessel  separately  at  the  same  temperature."  —  The  Steam 
Engine,  14th  ed.,  p.  220.  (6)  Engineering  Thermodynamics,  1907.  (7)  Op.  cit.,  I, 
104-107.  (8)  Op.  cit.,  593-595.  (9)  Nichols  and  Franklin,  The  Elements  of  Physics, 
I,  199-200.  (10)  Ibid.,  199;  Wormell,  Thermodynamics,  157-161.  (11)  Over  de 
Continuiteit  van  den  Gas  en  Vloeistoestand,  Leinden,  1873,  76  ;  tr.  by  Roth,  Leipsic, 
1887. 

SYNOPSIS   OF   CHAPTER   III 

Boyle's  law,  pv  =  PV:  deviations. 
Dalton's  law,  Avogadro's  principle. 
Law  of  Gay-Lussac  or  of  Charles  :  increase  of  volume  at  constant  pressure  ;  increase 

of  pressure  at  constant  volume  ;  values  of  the  coefficient  from  32°  F.  ;  deviations 

with  actual  gases. 


28  APPLIED  THERMODYNAMICS 

The  absolute  zero  :  —  459.4°  JP.,  or  491.4°  F.  below  the  freezing  point. 
Air  thermometers  :  Preston's  :  Hoadley's  :  calibration  :  gases  used. 

TJTT" 

The  perfect  gas,  ^—  —  — :  definitions :  properties  :  values  of  E  :  absence  of  inter- 
molecular  action  ;  the  kinetic  theory  ;  development  of  the  law  PV  =  ET  there- 
from ;  conformity  with  Avogadro's  principle  ;  molecular  velocity. 

The  Van  der  Waals  equation  for  imperfect  gases : 


PROBLEMS 

1.  Find  the  volume  of  one  pound  of  air  at  a  pressure  of  100  Ib.  per  square  inch, 
the  temperature  being  32°  F.,  using  Boyle's  law  only. 

2.  From  Charles'  law,  find  the  volume  of  one  pound  of  air  at  atmospheric  pres- 
sure and  72°  F. 

3.  Find  the  pressure  exerted  by  one  pound  of  air  having  a  volume  of  10  cubic 
feet  at  32°  F. 

4.  One  pound  of  air  is  cooled  from  atmospheric  pressure  at  constant  volume  from 
32°  F.  to  —  290°  F.     How  nearly  perfect  is  the  vacuum  produced  ? 

5.  Air  at  50  Ib.  per  square  inch  pressure  at  the  freezing  point  is  heated  at  con- 
stant volume  until  the  temperature  becomes  2900°  F.     Find  its  pressure  after  heating. 

6.  Five  pounds  of  air  occupy  50  cubic  feet  at  a  temperature  of  0°  F.     Find  the 
pressure. 

7.  Find  values  of  E  for  hydrogen,  nitrogen,  oxygen. 

8.  Find  the  volume  of  three  pounds  of  hydrogen  at  15  Ib.  pressure  per  square 
inch  and  75°  F. 

9.  Find  the  temperature  of  2  ounces  of  hydrogen  contained  in  a  1 -gallon  flask 
and  exerting  a  pressure  of  10,000  Ib.  per  square  inch. 

10.  Compute  the  value  of  r  (Art.  52). 

11.  Find  the  mean  molecular  velocity  of  1  Ib.  of  air  (considered  as  a  perfect  gas) 
at  atmospheric  pressure  and  70°  F. 

12.  How  large  a  flask  will  contain  1  Ib.  of  nitrogen   at    3200  Ib.  pressure   per 
square  inch  and  70°  F.  ? 


CHAPTER   IV 

THERMAL  CAPACITIES  :  SPECIFIC  HEATS  OF  GASES  :  JOULE'S  LAW 

57.  Thermal  Capacity.     The  definition  of  specific  heat  given  in  Art.  24  is, 
from  a  thermodynamic  standpoint,  inadequate.     Heat  produces  other  effects  than 
change  of  temperature.     A  definite  movement  of  heat  can  be  estimated  only  when 
all  of  these  effects  are  defined.     For  example,  the  quantity  of  heat  necessary  to 
raise  the  temperature  of  air  one  degree  in  a  constant  volume  air  thermometer  is 
much  less  than  that  used  in  raising  the  temperature  one  degree  in  the  constant 
pressure  type.     The  specific  heat  may  be  satisfactorily  defined  only  by  referring 
to  the  condition  of  the  substance  during  the  change  of  temperature.     Ordinary 
specific  heats  assume  constancy  of  pressure,  —  that  of  the  atmosphere,  —  while  the 
volume  increases  with  the  temperature  in  a  ratio  which  is  determined  by  the  coeffi- 
cient of  expansion  of  the  material.     A  specific  heat  determined  in  this  way  —  as 
are  those  of  solids  and  liquids  generally  —  is  the  specific  heat  at  constant  pressure. 

Whenever  the  term  "specific  heat"  is  used  without  qualification,  this  par- 
ticular specific  heat  is  intended.  Heat  may  be  absorbed  during  changes  of 
either  pressure,  volume,  or  temperature,  while  some  other  of  these  proper- 
ties of  the  substance  is  kept  constant.  For  a  specific  change  of  property, 
the  amount  of  heat  absorbed  represents  a  specific  thermal  capacity. 

58.  Expressions  for  Thermal  Capacities.   If  H  represents  heat  absorbed, 
c  a  constant  specific  heat,  and  (T—t)  a  range  of  temperature,  then,  by 

definition,   H=c(T—t)   and   c=H-*-(T—t).     If   c  be   variable,   then 

J'» 
cdT  and  c  =  dH-t-dT.     If  in  place  of  c  we  wish  to  denote  the 

specific  heat  at  constant  pressure  (A:),  or  that  at  constant  volume  (7),  we  may 
apply  subscripts  to  the  differential  coefficients  ;  thus, 

fdH\  fdH\ 

k  =  (^)     and  l=  77^0  ' 
\dTJp  \dTJr 

the  subscripts  denoting  the  property  which  remains  constant  during  the 
change  in  temperature. 

We  have  also  the  thermal  capacities, 


((1H\        (dH\       (djf\        fdH\ 
\dVlf    \dPJr'    \dPJr     \dVjp 


The  first  of  these  denotes  the  amount  of  heat  necessary  to  increase  the  specific 
volume  of  the  substance  by  unit  volume,  while  the  temperature  remains  constant  ; 

29 


30  APPLIED  THERMODYNAMICS 

this  is  known  as  the  latent  heat  of  expansion.  It  exemplifies  absorption  of  heat 
without  change  of  temperature.  No  names  have  been  assigned  for  the  other 
thermal  capacities,  but  uVis  not  difficult  to  describe  their  significance. 

59.  Values  of  Specific  Heats.    It  was  announced  by  Dulong  and  Petit  that  the 
specific  heats  of  substances  are  inversely  as  their  chemical  equivalents.     This  was 
shown  later  by  the  experiments  of  Regnault  and  others  to  be   approximately, 
though  not  absolutely,  correct.     Considering  metals  in  the  solid  state,  the  product 
of  the  specific  heat  by  the  atomic  weight  ranges  at  ordinary  temperatures  from  6.1 
to  6.5.     This  nearly  constant  product  is  called  the  atomic  heat.     Determination  of 
the  specific  heat  of  a  solid  metal,  therefore,  permits  of  the  approximate  computa- 
tion of  its  atomic  weight.     Certain  non-metallic  substances,  including  chlorine, 
bromine,  iodine,  selenium,  tellurium,  and  arsenic,  have  the  same  atomic  heat  as 
the  metals.     The  molecular  heats  of  compound  bodies  are  equal  to  the  sums  of  the 
atomic  heats  of  their  elements;  thus,  for  example,  for  common  salt,  the  specific 
heat  0.219,  multiplied  by  the  molecular  weight,  58.5,  gives  12.8  as  the  molecular 
heat;  which,  divided  by  2,  gives  6.4  as  the  average  atomic  heat  of  sodium  and 
chlorine ;  and  as  the  atomic  heat  of  sodium  is  known  to  be  6.4,  that  of  chlorine 
must  also  be  6.4  (1). 

60.  Volumetric  Specific   Heat.     Since  the  specific  volumes  of  gases  are  in- 
versely as  their  molecular  weights,  it  follows  from  Art.  59  that  the  quotient  of  the 
specific  heat  by  the  specific  volume  is  practically  constant  for  ordinary  gases.     In 
other  words,  the  specific  heats  of  equal  volumes  are  equal.     The  specific  heats  of 
these  gases  are  directly  proportional  to  their  specific  volumes  and  inversely  pro- 
portional to  their  densities,  approximately.     Hydrogen  must  obviously  possess  the 
highest  specific  heat  of  any  of  the  gases. 

61.  Mean,  "Real,"  and  "Apparent"  Specific  Heats.     Since  all  specific 
heats  are  variable,  the  values  given  in  tables  are  mean  values  ascertained 
over  a  definite  range  of  temperature.     The  mean  specific  heat,  adopting 
the  notation  of  Art.  58,  is  c  =  H-t-(T—  t);  while  the  true  specific  heat,  or 
specific  heat  "at  a  point,"  is  the  limiting  value  c  —  dff-r-  dT, 

Rankine  discusses  a  distinction  between  the  real  and  apparent  specific  heats ; 
meaning  by  the  former,  the  rate  of  heat  absorption  necessary  to  effect  changes  of 
temperature  alone,  without  the  performance  of  any  disgregation  or  external  work . 
and  by  the  latter,  the  observed  rate  of  heat  absorption,  effecting  the  same  change 
of  temperature,  but  simultaneously  causing  other  effects  as  well.  For  example, 
in  heating  water  at  constant  pressure  from  62°  to  63°  F.,  the  apparent  specific  heat 
is  1.0  (definition,  Art.  22).  To  compute  the  real  specific  heat,  we  must  know  the 
external  work  done  by  reason  of  expansion  against  the  constant  pressure,  and  the 
disgregation  work  which  has  readjusted  the  molecules.  Deducting  from  1.0 
the  heat  equivalent  to  these  two  amounts  of  work,  we  have  the  real  specific  heat, 
that  which  is  used  solely  for  making  the  substance  hotter.  Specific  heats  determined 
by  experiment  are  always  apparent;  the  real  specific  heats  are  known  only  by 
computation  (Art.  64). 


SPECIFIC  HEATS  OF  GASES  31 

62.  Specific  Heats  of  Gases.     Two  thermal  capacities  of  especial 
importance  are  used  in  calculations  relating  to  gases.     The  first  is 
the  specific  heat  at  constant  pressure,  k,  which  is  the  amount  of  heat 
necessary  to  raise  the  temperature  one  degree  while  the  pressure  is  kept 
constant;   the    other,  the    specific  heat   at   constant  volume,  1,  or  the 
amount  of  heat  necessary  to  raise  the  temperature  one  degree  while  the 
volume  is  kept  constant. 

63.  Regnault's  Law.      As  a  result  of  his  experiments  on  a  large  number  of 
gases  over  rather  limited  ranges  of  temperature,  Regnault   announced  that  the 
specific  heat  of  any  gas  at  constant  pressure  is  constant.     This  is  now  known  not  to 
be  rigorously  true  of  even  our  most  nearly  perfect  gases.     It  is  not  even  approxi- 
mately true  of  those  gases  when  far  from  the  condition  of  perfectness,  i.e.  at  low 
temperatures  or  high  pressures.     At  very  high  temperatures,  also,  it  is  well  known 
that  specific  heats  rapidly  increase.     This  particular  variation  is  perhaps  due  to 
an  approach  toward  that  change  of  state  described  as  dissociation.     When  near 
any  change  of  state,  —  combustion,  fusion,  evaporation,  dissociation, — every  sub- 
stance manifests  erratic  thermal  properties.     The  specific  heat  of  carbon  dioxide 
is  a  conspicuous  illustration.     Recent  determinations  by  Holborn  and  Henning 
(2)  of  the  mean  specific  heats  between  0°  and  x°  C.  give,  for  nitrogen,  k  =  0.255 
+  0.000019 a-;   and  for  carbon   dioxide,  k  =  0.201  +  0.0000742  x  -  0.00000001 8 re2: 
while  for  steam,  heated  from  110°  toz°C.,  £  =  0.4069-0.0000168  x  +  0.000000044  x\ 
The  specific  heats  of  solids  also  vary.     The  specific  heats  of  substances  in  general 
increase  with  the  temperature.     Regnault's  law  would  hold,  however,  for  a  perfect 
gas;  in  this,  the  specific  heat  would  be  constant  under  all  conditions  of  tempera- 
ture.    For  our  "permanent"  gases,  the  specific  heat  is  practically  constant  at 
ordinary  temperatures. 

64.  The  Two  Specific  Heats.  When  a  gas  is  heated  at  constant  pressure, 
its  volume  increases  against  that  pressure,  and  external  work  is  done  in 
consequence.  The  external  work  may  be  computed  by  multiplying  the 
pressure  by  the  change  in  volume.  When  heated  at  constant  volume,  no 
external  work  is  done ;  no  movement  is  made  against  an  external  resist- 
ance. If  the  gas  be  perfect,  then,  under  this  condition,  no  disgregation 
work  is  done ;  and  the  specific  heat  at  constant  volume  is  a  true  specific 
heat,  according  to  Rankine's  distinction  (Art.  61).  The  specific  heat  at 
constant  pressure  is,  however,  the  one  commonly  determined  by  experi- 
ment. The  numerical  values  of  the  two  specific  heats  must,  in  a  perfect 
gas,  differ  by  the  heat  equivalent  to  the  external  work  done  during  heating 
at  constant  pressure.  Under  certain  conditions,  —  as  with  water  at  its 
maximum,  density,  —  no  external  work  is  done  when  heating  at  constant 
pressure ;  and  at  this  state  the  two  specific  heats  are  equal,  if  we  ignore 
possible  differences  in  the  disgregation  work. 


32  APPLIED  THERMODYNAMICS 

65.  Difference  of  Specific  Heats.     Let  a  pound  of  air  at  32°  F. 
and  atmospheric  pressure  be  raised  1°  F.  in  temperature,  at  constant 
pressure.     It  will  expand  12.387 -j- 491.4  =  0.02521  cu.  ft.,  against 
a  resistance  of  14.7  x  144  =  2116.8  Ib.  per  square  foot.     The  external 
work  which  it  performs  is  consequently  2116.8  x  0.02521  =  53.36  foot- 
pounds.    A  general  expression  for  this  external  work  is  W=PV-s-  T; 
and  as  from  Art.  51  the  quotient  P  V-*-  T  is  a  constant  and  equal  to 
R,  then  TFis  a  constant  for  each  particular  gas,  and  equivalent  in 
value  to  that  of  R  for  such  gas.     The  value  of  W  for  air,  expressed 
in  heat  units,  is  53.36-1-778  =  0.0686.     If  the  specific  heat  of  air  at 
constant  pressure,  as  experimentally  determined,  be  taken  at  0.2375, 
then  the  specific  heat  at  constant  volume  is  0.2375  —  0.0686  =  0.1689, 
air  being  regarded  as  a  perfect  gas. 

66.  Derivation  of  Law  of  Perfect  Gas.     Let  a  gas  expand  at  constant  pres- 
sure P  from  the  condition  of  absolute  zero  to  any  other  condition  V,  T.    The  total 
external  work  which  it  will  have  done  in  consequence  of  this  expansion  is  PV. 
The  work  done  per  degree  of  temperature  is  PV---  T.     But,  by  Charles'  law,  this 
is  constant,  whence  we  have  PV=RT.     The  symbol  R  of  Art.  51  thus  represents 
the  external  work  of  expansion  during  each  degree  of  temperature  increase  (3). 

67.  General  Case.     The  difference  of  the  specific  heats,  while  constant  for  any 
gas,  is  different  for  different  gases,  because  their  values  of  R  differ.     But  since 
values  of  R  are  proportional  to  the  specific  volumes  of  gases  (Art.  52),  the  differ- 
ence of  the  volumetric  specific  heats  is  constant  for  all  gases.     Thus,  let  k,  I  be  the 
two  specific  heats,  per  pound,  of  air.     Then  k  —  I  =  r.     Let  d  be  the  density  of 
the  air;  then,  d(k—V)  is  the  difference  of  the  volumetric  specific  heats.     For  any 
other  gas,  we  have   similarly,  K  —  L  =  R   and   D(K  —  L)  ;   but,  from  Art.   52 
R:r::d:D,    or    R  =  rd  -  D.      Hence,    K  -  L  =  rd  +  D  =  (k  -  l}(d  -4-  D),    or 
D(K  —  L}  =  d(k  —  /).     The  difference  of  the  volumetric  specific  heats  is  for  all 
gases  equal  approximately  to  0.0055  B.  t.  u.     (Compare  Art.  60.) 

68.  Computation  of  External  Work.     The  value  of  R  given  in   Art.  52  and 
Art.  65  is  variously  stated  by  the   writers   on   the    subject,    on  account  of  the 
slight  uncertainty  which  exists  regarding  the  exact  values  of  some  of  the  con- 
stants used  in  computing  it.     The  differences  are  too  small  to  be  of  consequence 
in  engineering  work. 

69.  Ratio  of  Specific  Heats.     The  numerical  ratio   between    the 
two  specific  heats  of  a  sensibly  perfect  gas,  denoted  by  the  symbol  y, 
is  a  constant  of  prime  importance  in  thermodynamics. 

For  air,  its  value  is  0.2375  H-  0.1689  =  1.402.     Various  writers,  using  other 
fundamental  data,  give  slightly  different  values  (4).     The  best  direct  experiments 


SPECIFIC  HEATS   OF   GASES  33 

(to  be  described  later)  agree  with  that  here  given  within  a  narrow  margin.  For 
hydrogen,  Luinmer  and  Pringsheim  (5)  have  obtained  the  value  1.408 ;  and  for 
oxygen,  1.396.  For  carbon  dioxide,  a  much  less  perfect  gas  than  any  of  these, 
these  observers  make  the  value  of  y,  1.2961;  while  Dulong  obtained  1.338.  The 
latter  obtained  for  carbon  monoxide  1.428.  The  mean  value  for  the  "  permanent " 
gases  is  close  to  that  for  air,  viz., 

y  =  1.402. 

The  value  should  be  the  same  for  all  gases  as  they  closely  approach 
perfectness;  for  as  the  law  PV=  RT  holds,  so  must  the  difference  of 
the  specific  heats  be  absolutely  constant ;  and  as  Regnault's  law  (Art.  63) 
holds,  the  two  specific  heats  must  themselves  be  constant ;  whence  their 
ratio  must  also  be  constant.  The  value  of  the  ratio  for  ordinary  actual 
gases  is  independent  of  the  temperature  and  the  pressure. 

70.  Relations  of  R  and  y.  A  direct  series  of  relations  exists 
between  the  two  specific  heats,  their  ratio,  and  their  difference.  If 
we  denote  the  specific  heats  by  k  and  ?,  then  in  proper  units, 


f For  air,  this  gives  °-237f.   Qfi  =  1.402.) 

V  n  <nr»rrr          OO.OD  / 


0.2375  -^p 

778 


I  =  R--.  R  ==  k?  =  l(y  -  1). 

y-1  y 

71.  Ranklne's  Prediction  of  the  Specific  Heat  of  Air.     The  specific  heat  of  air 
was  approximately  determined   by  Joule  in  1852.     Earlier  determinations  were 
unreliable.     Rankine,  in  1850,  by  the  use  of  the  relations  just  cited,  closely  ap- 
proximated the  result  obtained  experimentally  by  Regnault  three  years   later. 
Using  the  values  y  =  1.4,  R  =  53.15,  Rankine  obtained 

*  =  R    V     =  (53.15  -s-  772)  x  (1.4  -*•  0.4)  =  0.239. 

y-  i 

Regnault's  result  was  0.2375. 

72.  Mayer's  Computation  of  the  Mechanical  Equivalent  of  Heat. 
Reference  was  made  in   Art.  29  to  tlie  computation  of  tins  constant 
prior  to  tbe  date  of  Joule's  conclusive  experiments.     The  method  is 
substantially  as  follows  :  a  cylinder  and  piston  having  an  area  of  one 
square  foot,  the  former  containing  one  cubic  foot,  are  assumed  to  hold 


34  APPLIED  THERMODYNAMICS 

air  at  32°  F.,  which  is  subjected  to  heat.  The  piston  is  balanced,  so 
that  the  pressure  on  the  air  is  that  of  the  atmosphere,  or  14.7  Ib. 
per  square  inch ;  the  total  pressure  on  the  piston  being,  then, 
144  x  14.7  =  2116.8  Ib.  While  under  this  pressure,  the  air  is  heated 
until  its  temperature  has  increased  491.4°.  The  initial  volume 
of  the  air  was  by  assumption  one  cubic  foot,  whence  its  weight 
was  1  -r- 12.387  =  0.0811  Ib.  The  heat  imparted  was  therefore 
0.0811  x  0.2375  x  491.4  =  9.465  B.  t.  u.  The  external  work  was 
that  due  to  doubling  the  volume  of  the  air,  or  1  x  14.7  x  144  =  2116.8 
foot-pounds.  The  piston  is  now  fixed  rigidly  in  its  original  position, 
so  that  the  volume  cannot  change,  and  no  external  work  can  be  done. 
The  heat  required  to  produce  an  elevation  of  temperature  of  491.4° 
is  then  0.0811x0.1689x491.4  =  6.731  B.  t.  u.  The  difference 
of  heat  corresponding  to  the  external  work  done  is  2.734  B.  t.  u., 
whence  the  mechanical  equivalent  of  heat  is  2116.8  -T-  2.734=  774.2 
foot-pounds. 

73,  Joule's  Experiment.  One  of  the  crucial  experiments  of  the  science  was 
conducted  by  Joule  about  1844,  after  having  been  previously  attempted  by  Gay- 
Lussac. 

Two  copper  receivers,  A  and  J5,  Fig.  9,  were  connected  by  a  tube 
and  stopcock,  and  placed  in  a  water  bath.  Air  was  compressed  in  A 

to  a  pressure  of  22  atmospheres, 
while  a  vacuum  was  maintained 
in  B.  When  the  stopcock  was 
opened,  the  pressure  in  each  re- 
ceiver became  11  atmospheres,  and 
the  temperature  of  the  air  and  of 


FIG.  9.    Arts.  73,  80.  —  Joule's  Experiment. 

the  water  bath  remained  practically 

unchanged.  This  was  an  instance  of  expansion  without  the  perform- 
ance of  external  work  ;  for  there  was  no  resisting  pressure  against  the 
augmentation  of  volume  of  the  air. 

74.  Joule's  and  Kelvin's  Porous  Plug  Experiment.  Minute  observations 
showed  that  a  slight  change  of  temperature  occurred  in  the  water  bath. 
Joule  and  Kelvin,  in  1852,  by  their  celebrated  "porous  plug"  experiments, 
ascertained  the  exact  amount  of  this  change  for  various  gases.  In  all  of 
the  permanent  gases  the  change  was  very  small ;  in  some  cases  the  tern- 


JOULE'S   LAW  35 

perature  increased,  while  in  others  it  decreased ;  and  the  inference  is  jus- 
tified that  in  a  perfect  gas  there  would  be  no  change  of  temperature  (Art. 
156). 

75.  Joule's  Law.     The   experiments   led   to   the  principle   that 
when  a  perfect  gas  expands  without  doing  external  work,  and  without 
receiving  or  discharging  heat,  the  temperature  remains  unchanged  and 
no  disgregation  work  is  done.     A  clear  appreciation  of  this  law  is  of 
fundamental   importance.     Four   thermal    phenomena    might   have 
occurred  in  Joule's  experiment :  a  movement  of  heat,  the  performance 
of  external  work,  a  change  in  temperature,  or  work  of  disgregation. 
From  Art.  12,  these  four  effects  are  related  to  one  another  in  such 
manner  that  their  summation  is  zero;   (J2"=  T+ 1+  W).     By  means 
of  the  water  bath,  which  throughout  the  experiment  had  the  same 
temperature  as  the  air,  the  movement  of  heat  to  or  from  the  air  was 
prevented.     By  expanding  into  a  vacuum,  the  performance  of  external 
work  was  prevented.     The  two  remaining  items  must  then  sum  up 
to  zero,  i.e.  the  temperature  change  and  the  disgregation  work.     But 
the  temperature  did  not  change  ;  consequently  the  amount  of  disgre- 
gation work  must  have  been  zero. 

76.  Consequences  of  Joule's  Law.    In  the  experiment  described,  the  pres- 
sure and  volume  changed  without  changing  the  internal  energy.     No  dis- 
gregation work   was   done,   and  the   temperature   remained   unchanged. 
Considering  pressure,  volume,  and  temperature  as  three  cardinal  thermal 
properties,  internal  energy  is  then  independent  of  the  pressure  or  volume 
and  depends  on  the  temperature  only,  in  any  perfect  gas.     It  is  thus  itself 
a  cardinal  property,  in  this  case,  a  function  of  the  temperature.      "A 
change  of  pressure  and  volume  of  a  perfect  gas  not  associated  with  change 
of  temperature  does  not  alter  the  internal  energy.     In  any  change  of  tem- 
perature, the  change  of  internal  energy  is  independent  of  the  relation  of 
pressure  to  volume  during  the  operation ;  it  depends  only  on  the  amount 
by  which  the  temperature  has  been  changed"  (6).     The  gas  tends  to  cool 
in  expanding,  but  this  effect  is  "  exactly  compensated  by  the  heat  which 
is  disengaged  through  the  friction  in  the  connecting  tube  and  the  im- 
pacts which  destroy  the  velocities  communicated  to  the  particles  of  gas 
while^  it  is  expanding"  (7).     Tliere  is  practically  no  disgregation  ivork  in 
heating  a  sensibly  perfect  gas;   all  of  the  internal  energy  is  evidenced  by 
temperature  alone.     When  such  a  gas  passes  from  one  state  to  another  in 
a  variety  of  ways,  the  external  work  done  varies;  but  if  from  the  total 


36  APPLIED  THERMODYNAMICS 

movement  of  heat  the  equivalent  of  the  external  work  be  deducted,  then 
the  remainder  is  always  the  same,  no  matter  in  what  way  the  change  of 
condition  has  been  produced.  Instead  of  H  =  T  +  /  -f  JF,  we  may  write 
H=  T+  W. 

77.  Application  to  Difference  of  Specific  Heats.     The  heat  absorbed  dur- 
ing a  change  in  temperature  at  constant  pressure  being  H=  k(T—  t\  and 
the  external  work  during  such  a  change  being  W=  P(V—  v)  =  R(T—  t), 
the  gain  of  internal  energy  must  be 

H-  W=(k-r)(T-t). 

The  heat  absorbed  during  the  same  change  of  temperature  at  constant 
volume  is  H—l(T—t).  Since  in  this  case  no  external  work  is  done,  the 
whole  of  the  heat  must  have  been  applied  to  increasing  the  internal  energy. 
But,  according  to  Joule's  law,  the  change  of  internal  energy  is  shown  by  the 
temperature  change  alone.  In  whatever  way  the  temperature  is  changed 
from  T  to  t,  the  gain  of  internal  energy  is  the  same.  Consequently, 

(k-R)(T-t)=l(T-f)  and  k-R  =  l, 
a  result  already  suggested  in  Art.  65. 

78.  Discussion  of  Results.     The  greater  value  of  the  specific  heat  at 
constant  pressure  is  due  solely  to  the  performance  of  external  work  dur- 
ing the  change  in  temperature.     The  specific  heat  at  constant  volume  is 
a  real  specific  heat,  in  the  case  of  a  perfect  gas ;  no  external  work  is  done, 
and  the  internal  energy  is  increased  only  by  reason  of  an  elevation  of  tem- 
perature.    There  is  no  disgregation  work.     All  of  the  heat  goes  to  make 
the  substance  hot.     So  long  as  no  external  work  is  done,  it  is  not  neces- 
sary to  keep  the  gas  at  constant  volume  in  order  to  confirm   the  lower 
value  for  the  specific  heat;  no  more  heat  is  required  to  raise  the  tempera- 
ture a  given  amount  when  the  gas  is  allowed  to  expand  than  when  the 
volume  is  maintained  constant.     For  any  gas  in  which  the  specific  heat  at 
constant  volume  is  constant,  Joule's  law  is  inductively  established  ;  for  no 
external  work  is  done,  and  temperature  alone  measures  the  heat  absorp- 
tion at  any  point  on  the  thermometric  scale.    If  a  gas  is  allowed  to  expand, 
doing  external  work  at  constant  temperature,  then,  since  no  change  of  inter- 
nal energy  occurs,  it  is  obvious  from  Art.  12  that  the  external  ivork  is  equal 
to  the  heat  absorbed.     Briefly,  the  important  deduction  from  Joule's  experi- 
ment is  that  item  (6),  Art.  12,  may  be  ignored  when  dealing  with  sensibly 
perfect  gases. 

79.  Confirmatory  Experiment.     By    a    subsequent    experiment,    Joule 
showed  that  when  a  gas  expands  so  as  to  perform  external  work,  heat  dis- 


JOULE'S  LAW  37 

appears  to  an  extent  proportional  to  the  work  done.  Figure  10  illustrates 
the  apparatus.  A  receiver  A,  containing  gas  compressed  to  two  atmos- 
pheres, was  placed  in  the  calorimeter  B  and  connected  with  the  gas  holder 
(7,  placed  over  a  water  tank.  The  gas  passed 
from  A  to  C  through  the  coil  Z>,  depressed  the 
water  in  the  gas  holder,  and  divided  itself  be- 
tween the  two  vessels,  the  pressure  falling  to 
that  of  one  atmosphere.  The  work  done  was 

computed  from  the  augmentation  of  volume  shown  FlG'  10>  Art<  79-— Joule's 
,  j  .  .  ,  , ,  „  .  Experiment,  Second  Ap- 

by  driving  down  the  water  in  C  against  atmos-       paratus 

pheric  pressure ;  and  the  heat  lost  was  ascertained 

from  the  fall  of  temperature  of  the  water.  If  the  temperature  of  the 
air  were  caused  to  remain  constant  throughout  the  experiment,  then  the 
work  done  at  C  would  be  precisely  equivalent  to  the  heat  given  up  by 
the  water.  If  the  temperature  of  the  air  were  caused  to  remain  constantly 
the  same  as  that  of  the  water,  then  H=Q=T+  /+  W,  (T+  /)=  -  W,  or 
internal  energy  would  be  given  up  by  the  air,  precisely  equivalent  in  amount 
to  the  work  done  in  C. 

80-  Application  of  the  Kinetic  Theory.  In  the  porous  plug  experiment  referred 
to  in  Art.  74,  it  was  found  that  certain  gases  were  slightly  cooled  as  a  result  of  the 
expansion,  and  others  slightly  warmed.  The  molecules  of  gas  are  very  much  closer 
to  one  another  in  A  than  in  B,  at  the  beginning  of  the  experiment.  If  the  mole- 
cules are  mutually  attractive,  the  following  action  takes  place :  as  they  emerge  from 
A,  they  are  attracted  by  the  remaining  particles  in  that  vessel,  and  their  velocity 
decreases.  As  they  enter  B,  they  encounter  attractions  there,  which  tend  to  in- 
crease their  velocity;  but  as  the  second  set  of  attractions  is  feebler,  the  total  effect 
is  a  loss  of  velocity  and  a  cooling  of  the  gas.  In  another  gas,  in  which  the  molecules 
repel  one  another,  the  velocity  during  passage  would  be  on  the  whole  augmented, 
and  the  temperature  increased.  A  perfect  gas  would  undergo  neither  increase  nor 
decrease  of  temperature,  for  there  would  be  no  attractions  or  repulsions  between 
the  molecules. 

(1)  A  critical  review  of  this  theory  has  been  presented  by  Mills  :  The  Specific 
Heats  of  the  Elements,  Science,  Aug.  24,  1908,  p.  221.  (2)  The  Engineer,  January 
13,  1908.  (3)  Throughout  this  study,  no  attention  will  be  paid  to  the  ratio  778  as 
affecting  the  numerical  value  of  constants  in  formulas  involving  both  heat  arid  work 
quantities.  The  student  should  discern  whether  heat  units  or  foot-pounds  are  in- 
tended. (4)  Zeuner,  Technical  Thermodynamics,  Klein  tr.,  I,  121.  (5)  Ibid.,  loc. 
cit.  (6)  Ewing :  The  Steam  Engine,  1906.  (7)  Wormell,  Thermodynamics. 


SYNOPSIS   OF   CHAPTER   IV 

Specific  thermal  capacities  ;  at  constant  pressure,  at  constant  volume:  other  capacities. 

Atomic  heat  =  specific  heat  x  atomic  weight;  molecular  heat. 

The  volumetric  specific  heats  of  common  gases  are  approximately  equal. 


38  APPLIED  THERMODYNAMICS 

rr  (Iff 

Mean  specific  heat  = ;  true  specific  heat  =—=,;  real  and  apparent  specific  heats. 

T  —  t  dT 

BegnaulVs  law :  "  the  specific  heat  is  constant  for  perfect  gases." 

Difference  of  the  two  specific  heats  :  B  =  53.36  ;  significance  of  B. 

The  difference  of  the  volumetric  specific  heats  equals  0.0055  B.  t.  u.  for  all  gases. 

Ratio  of  the  specific  heats  :  y  =  1.402  for  air  ;  relations  between  k,  Z,  y,  B. 

Rankine's  prediction  of  the  value  of  k :  Mayer's  computation  of  the  mechanical  equiva- 
lent of  heat. 

Joule'1  s  Law :  no  disgregation  work  occurs  in  a  perfect  gas. 

If  the  temperature  does  not  change,  the  external  work  equals  the  heat  absorbed. 

If  no  heat  is  received,  internal  energy  disappears  to  an  extent  equivalent  to  the 
external  work  done. 

The  condition  of  intermolecular  force  determines  whether  a  rise  or  a  fall  of  temperature 
occurs  in  the  porous  plug  experiment. 


PROBLEMS 

1.  The  atomic  weights  of  iron,  lead,  and  zinc  being  respectively  56,  206.4,  65;  and 
the  specific  heats  being,  for  cast  iron,  0.1298;  for  wrought  iron,  0.1138;  for  lead, 
0.0314  ;  and  for  zinc,  0.0956,  —  check  the  theory  of  Art.  59  and  comment  on  the  results. 

2.  Find  the  volumetric  specific  heats  at  constant  pressure  of  air,  hydrogen,  arid 
nitrogen,  and  compare  with  Art.  60.     (k  =  3.4  for  H  and  0.2438  for  JV.) 

3.  The  heat  expended  in  warming  water  from  32°  F.  to  160°  F.  being  127.86  B.  t.  u. , 
find  the  mean  specific  heat  over  this  range. 

4.  The  weight  of  a  cubic  foot  of  water  being  59.83  Ib.  at  212°  F.  and  62.422  Ib.  at 
32°  F.,  find  the  amount  of  heat  expended  in  performing  external  work  when  one  pound 
of  water  is  heated  between  these  temperatures  at  atmospheric  pressure. 

5.  (a)  Find  the  specific  heat  at  constant  volume  of  hydrogen  and  nitrogen. 
(5)  Find  the  value  of  y  for  these  two  gases. 

6.  Check  the  value  0.0055  B.  t.  u.  given  in  Art.  67  for  hydrogen  and  nitrogen. 

7.  Compute  the  elevation  in  temperature,  in  Art.  72,  that  would,  for  an  expansion 
of  100  per  cent,  under  the  assumed  conditions,  and  with  the  given  values  of  k  and  Z, 
give  exactly  778  as  the  value  of  the  mechanical  equivalent  of  heat.     What  law  of 
gaseous  expansion  would  be  invalidated  if  this  elevation  of  temperature  occurred  ? 

8.  In  the  experiment  of  Art.  79,  the  volume  of  air  in  C  increased  by  one  cubic  foot 
against  normal  atmospheric  pressure.     The  weight  of  water  in  B  was  20  Ib.     The  tem- 
perature of  the  air  remained  constant  throughout  the  experiment.    Ignoring  radiation 
losses,  compute  the  fall  of  temperature  of  the  water. 

9.  Prove  that  the  specific  heat  at  constant  pressure  is  constant  for  a  perfect  gas. 


CHAPTER  V 

GRAPHICAL  REPRESENTATIONS:  PRESSURE- VOLUME  PATHS  OF 

PERFECT  GASES 

81.  Thermodynamic  Coordinates.     The  condition  of  a  body  being  fully 
defined  by  its  pressure,  volume,  and  temperature,  its  state  may  be  repre- 
sented on  a  geometrical  diagram  in  which  these  properties  are  used  as 
coordinates.     This  graphical  method  of  analysis,  developed  by  Clapeyron, 
is  now  in  universal  use.     The  necessity  for  three  coordinates  presupposes 
the  use  of  analytical  geometry  of  three  dimensions,  and  representations 
may  then  be  shown  perspectively  as  related  to  one  of  the  eight  corners 
of  a  cube;  but  the  projections  on  any  of  the  three  adjacent  cube  faces  are 
commonly  used ;  and  since  any  two  of  three  properties  fix  the  third  when 
the  characteristic  equation  is  known,  a  projective  representation  is  suffi- 
cient.    Since  internal  energy  is  a  cardinal  property  (Arts.  10,  76),  this  also 
may  be  employed  as  one  of  the  coordinates  of  a  diagram  if  desired. 

82.  Illustration.     In  Fig.   11  we   have  one   corner  of  a   cube 
constituting  an  origin  of  coordinates  at  O.     The  temperature  of  a 
substance  is  to  be  represented  by  the  distance  upward  from  0;  its 
pressure,  by  the  distance. to  the  right;  and  its  volume,  by  the  dis- 
tance to  the  left.     The  lines  forming  the  cube  edges  are  correspond- 
ingly marked  OT,  OP,  0V.     Consider  the  condition  of  the  body  to 
be  represented  by  the  point  A,  within  the  cube.     Its  temperature  is 
then  represented  by  the  distance  AB,  parallel  to  TO,  the  point  B 
being  in  the  plane  VOP.     The  distance  AD,  parallel  to  P  0,  from  A 
to  the  plane  TOV,  indicates  the  pressure;  and  by  drawing  AC  paral- 
lel to  VO,  C  being  the  intersection  of  this  line  with  the  plane  TOP, 
we  may  represent  the  volume.     The  state  of  the  substance  is  thus 
fully  shown.    Any  of  the  three  projections,  Figs.  12-14,  would  equally 
fix  its  condition,  providing  the  relation  between  P,    V,   and   T  is 
known.     In  each  of  these  'projections,  two  of  the  properties  of  the 
substance  are  shown  ;  in  the  three  projections,  each  property  appears 

39 


40 


APPLIED  THERMODYNAMICS 


twice;  and  the  corresponding  lines  AB,  AC,  and  AD  are  always 
equal  in  length. 


\ 

1 
FIG.  11. 

Persp 
gram. 

A 

^\c 

/r                             D 

/                                0 

3 

Art.  82.  —        FIG 
ective    Dia- 

V 

A.B, 

!                       D 

T 
C—  -j 

o'  —  s  v 

0 
p                     <~l 

B                                              C 
.  12.    Art.  82.—        FIG.  13.    Art.  82.— 
TP  Diagram.                  VP  Diagram. 

w        B 
FIG.  14.    Art.  82.— 
TV  Diagram. 

83.  Thermal  Lines.  In  Fig.  15,  let  a  substance,  originally  at  A,  pass 
at  constant  pressure  and  temperature  to  the  state  B]  thence  at  constant 
temperature  and  volume  to  the  state  C;  and  thence  at  constant  pressure 

T 

V 


B,C, 


FIG.  15.  Art.  83.— 
Perspective  Ther- 
mal Line. 


FIG.  16.    Art.  83.  — 
TP  Path. 


FIG.  17.    Art.  83.— 
VP  Path. 


FIG.  18.    Art.  83. 
TV  Path. 


and  volume  to  D.  Its  changes  are  represented  by  the  broken  line  ABCD, 
which  is  shown  in  its  various  projections  in  Figs.  16-18.  The  thermal 
line  of  the  coordinate  diagrams,  Figs.  11  and  15,  is  the  locus  of  a  series  of 
successive  states  of  the  substance.  A  path  is  the  projection  of  a  thermal 
line  on  one  of  the  coordinate  planes  (Figs.  12-14,  16-18).  The  path  of  a 
substance  is  sometimes  called  its  process  curve,  and  its  thermal  line,  a 
thermogram. 

The  following  thermal  lines  are  more  or  less  commonly  studied :  — 

(a)  Isothermal,  in  which  the  temperature  is  constant;  its  plane  is 
perpendicular  to  the  OT  axis. 

(5)  Isometric,  in  which  the  volume  is  constant ;  having  its  plane  per- 
pendicular to  the  0V axis. 

(c)  Isopiestic,  in  which  the  pressure  is  constant ;  its  plane  being  per- 

pendicular to  the  OP  axis. 

(d)  Isodynamic,  that  along  which   no  change   of   internal   energy 

occurs. 


GRAPHICAL  REPRESENTATIONS 


41 


(e)  Adiabatic,  that  along  which  no  heat  is  transferred  between  the 
substance  and  surrounding  bodies;  the  thermal  line  of  an 
insulated  body. 

84.  Thermodynamic  Surface.  Since  the  equation  of  a  gas  in- 
cludes three  variables,  its  geometrical  representation  is  a  surface ; 
and  the  first  three,  at  least,  of  the  above  paths,  must  be  projections 
of  the  intersection  of  a  plane  with  such  surface.  Figure  19,  from  Pea- 


FIG.  19.    Arts.  £4,  103.— Thermodynamic  Surface  for  a  Perfect  Gas. 


body  (1),  admirably  illustrates  the  equation  of  a  perfect  gas,  PV= 
RT.  The  surface  pmnv  is  the  characteristic  surface  for  a  perfect  gas. 
Every  section  of  this  surface  parallel  to  the  PV plane  is  an  equilat- 
eral hyperbola.  Every  projection  of  such  section  on  the  P  V  plane 
is  also  an  equilateral  hyperbola,  the  coordinates  of  which  express  the 
law  of  Boyle,  PV—C.  Every  section  parallel  with  the  TV  plane 
gives  straight  lines  pm,  si,  etc.,  and  every  section  parallel  with  the 
TP  plane  gives  straight  lines  vn,  xy,  etc.  The  equations  of  these 


42 


APPLIED  THERMODYNAMICS 


FIG.  20.    Art.  85.  — Water 
at  Constant  Pressure. 


lines  are  expressions  of  the  two  forms  of  the  law  of  Charles,  their 
appearance  being  comparable  with  that  in  Fig.  5. 

85.  Path  of  Water  at  Constant  Pressure.     Some  such  diagram  as  that 
of  Fig.  20  would  represent  the  behavior  of  water  in  its  solid,  liquid,  and 

vaporous  forms  when  heated  at  constant  pressure. 
The  coordinates  are  temperature  and  volume.  At 
A,  the  substance  is  ice,  at  a  temperature  below 
the  freezing  point.  As  the  ice  is  heated  from  A 
to  B,  it  undergoes  a  slight  expansion,  like  other 
solids.  At  B,  the  melting  point  is  reached,  and 
as  ice  contracts  in  melting,  there  is  a  decrease  in 
volume  at  constant  temperature.  At  (7,  the  sub- 
stance is  all  water;  it  contracts  until  it  reaches  the 
temperature  of  maximum  density,  39.1°  F.,  at  Z>, 
then  expands  until  it  boils  at  E,  when  the  great 

increase  in  volume  of  steam  over  water  is  shown  by  the  line  EF.  If  the 
steam  after  formation  conformed  to  Charles'  law,  the  path  would  con- 
tinue upward  and  to  the  right  from  F,  as  a  straight  line. 

86.  The  Diagram  of  Energy.     Of  the  three  coordinate  planes,  the  PV 
is  most  commonly  used.     This  gives  a  diagram  corresponding  with  that 
produced  by  the  steam  engine  indicator  (Art.  484).    It  is  sometimes  called 
Watts'  diagram.     Its  importance  arises  principally  from  the  fact  that  it 
represents  directly  the  external  work  done  during  the  movement  of  the 
substance  along  any  path.     Consider  a  vertical  cylinder  filled  with  fluid, 
at  the  upper  end  of  which  is  placed  a  weighted  piston.     Let  the  piston  be 
caused  to  rise  by  the  expansion  of  the  fluid.     The  force  exerted  is  then 
equivalent  to  the  weight  of  the  piston,  or  total  pressure  on  the  fluid ;  the 
distance  moved  is  the  movement  of  the  piston,  which  is  equal  to  the  aug- 
mentation in  volume  of  the  fluid.     Since  work  equals  force  multiplied  by 
distance  moved,  the  external  work  done  is  equal  to  the  total  uniform  pressure 
multiplied  by  the  increase  of  volume. 

87.  Theorem.      On  a  PV  diagram,  the  external  work  done  along 
any  path  is  represented  by  the  area  included  be-      p 

tween  that  path  and  the   perpendiculars  from  its 
extremities  to  the  horizontal  axis. 

Consider  first  a  path  of  constant  pressure,  ab, 
Fig.  21.  From  Art.  86,  the  external  work  is 
equivalent  to  the  pressure  multiplied  by  the  in-  FlG-  21-  Art-  87-~ 

J  External  Work  at 

crease  ot  volume,  or  to  ca  x  ab  =  caod.      General      Constant  Pressure. 


CYCLES 


43 


case :  let  the  path  be  arbitrary,  ah,  Fig.  22.  Divide  the  area  abdc 
into  an  infinite  number  of  vertical  strips,  amnc,  mopn,  oqrp,  etc., 
each  of  which  may  be  regarded  as  a  rectangle, 
such  that  ac  =  mn,  mn  =  op,  etc.  The  external 
work  done  along  am,  mo,  oq,  etc.,  is  then  repre- 
sented by  the  areas  amnc,  mopn,  oqrp,  etc.,  and 
the  total  external  work  along  the  path  ab  is  repre- 
sented by  the  sum  of  these  areas,  or  by  abdc.  c  n  p  r  d 

FIG.  22.    Arts.  87, 88. 

Corollary  I.  Along  a  path  of  constant  volume  —External  Work, 
no  external  work  is  done. 

Corollary  II.  If  the  path  be  reversed,  i.e.  from  right  to  left,  as 
along  ba,  the  volume  is  diminished,  and  negative  ivork  is  done  ;  work 
is  expended  on  the  substance  in  compressing  it,  instead  of  being  per- 
formed by  it. 

88.  Significance  of  Path.     It  is  obvious,  from  Fig.  22,  that  the  amount 
of  external  work  done  depends  not  only  on  the  initial  and  final  states  a  and 
&,  but  also  on  the  nature  of  the  path  between  those  states.     According  to 
Joule's  principle  (Art.  75)  the  change  of  internal  energy  (T+I,  Art.  12) 
between  two  states  of  a  perfect  gas  is  dependent  upon  the  initial  and  final 
temperatures  only  and  is  independent  of  the  path.     The  external  work 
done,  however,  depends  upon  the  path.     The  total  expenditure  of  heat,  which 
includes  both  effects,  can  only  be  known  when  the  path  is  given.     The 
internal  energy  of  a  perfect  gas  (and,  as  will  presently  be  shown,  Art. 
109,  of  any  substance)  is  a  cardinal  property ;  external  work  and  heat 
transferred  are  not.     They  cannot  be  used  as  elements  of  a  coordinate 
diagram. 

89.  Cycle.     A  series  of  paths  forming  a  closed  finite  figure  con- 

stitutes a  cycle.  In  a  cycle,  the  substance  is  brought 
back  to  its  initial  conditions  of  pressure,  volume, 
and  temperature. 

Theorem.  In  a  cycle,  the  net  external  work 
done  is  represented  on  the  PV  diagram  by  the  en- 
closed area. 

Let  abed,  Fig.  23,  be  any  cycle.  Along  abc,  the 
work  done  is,  from  Art.  87,  represented  by  the 
area  abcef.  Along  cda,  the  negative  work  done  is  similarly  repre- 


FIG.  23.  Art.  89. — 
External  Work  in 
Closed  Cycle. 


44  APPLIED  THERMODYNAMICS 

sented  by  the  area  adcef.  The  net  positive  work  done  is  equivalent 
to  the  difference  of  these  two  areas,  or  to  abed. 

If  the  volume  units  are  in  cubic  feet,  and  the  pressure  units  are  pounds 
per  square  foot,  then  the  measured  area  abed  gives  the  work  in  foot-pounds. 
This  principle  underlies  the  calculation  of  the  horse  power  of  an  engine 
from  its  indicator  diagram.  If  the  cycle  be  worked  in  a  negative  direction, 
e.g.  as  cbad,  Fig.  23,  then  the  net  work  will  be  negative ;  i.e.  work  will 
have  been  expended  upon  the  substance,  adding  heat  to  it,  as  in  an  air 
compressor. 

90.  Theorem.     In  a  perfect  gas  cycle,  the  expenditure  of  heat  is 
equivalent  to  the  external  work  done. 

Since  the  substance  has  been  brought  back  to  its  initial  tempera- 
ture, and  since  the  internal  energy  depends  solely  upon  the  tempera- 
ture, the  only  heat  effect  is  the  external  work.  In  the  equation 
ff=  T+I+  W,  T+I=  0,  whence  H=  W,  the  expenditure  of  heat 
being  equivalent  to  its  sole  effect.* 

If  the  work  is  measured  in  foot-pounds,  the  heat  expended  is  calcu- 
lated by  dividing  by  778.  (See  Note  3,  page  37.)  Conversely,  in  a 
reversed  cycle,  the  expenditure  of  external  work  is  equivalent  to  the  gain  of 
heat. 

91.  Isothermal  Expansion.     The  isothermal  path  is  one  of  much 
importance  in  establishing  fundamental  principles.     By  definition 
(Art.  83)  it  is  that  path  along  which  the  temperature  of  the  fluid 
is  constant.     For  gases,  therefore,  from  the  characteristic  equation, 
if  T  be  made  constant,  the  isothermal  equation  is 

PV=RT=C. 

Taking  R  at  53.36  and  T  at  491.4°  (32°  F.),  0=  53.36  x  491.4  = 
26,221.104 ;  whence  we  plot  on  Fig.  2  the  isothermal  curve  ab  for 
this  temperature ;  an  equilateral  hyperbola,  asymptotic  to  the  axes 
of  P  and  V.  An  infinite  number  of  isothermals  might  be  plotted, 
depending  upon  the  temperature  assigned,  as  cd,  ef,  gh,  etc.  The 
equation  of  the  isothermal  may  be  regarded  as  a  special  form  of  the 
exponential  equation  PVn  —  C,  in  which  n  —  \. 

*  It  may  be  inferred  later  (Art.  109)  that  this  theorem  is  valid  for  substances  in 
general. 


ISOTHERMAL  EXPANSION 


45 


92.  Graphical  Method.  For  rapidly  plotting  curves  of  the  forrii  PV  -  C,  the 
construction  shown  in  Fig.  24  is  useful.  Knowing  the  three  corresponding  prop- 
erties of  the  gas  at  any  given  o 
state  enables  us  to  fix  one  point 
on  the  curve ;  thus  the  volume 
12.387  and  the  pressure  2116.8 
give  us  the  point  C  on  the 
isothermal  for  491.4°  absolute. 
Through  C  draw  CM  parallel 
to  0  V.  From  0  draw  lines  OD, 
ON,  OM  to  meet  CM.  Draw 
CB  parallel  to  OP.  From  th3 
points  1,  5,  6,  where  OD,  ON, 


FIG.  24.    Art.  92,  95.  — Construction    of   Equilateral 
Hyperbola. 


OM  intersect    CB,  draw  lines 

1  2,  5  7,  6  8  parallel  to  OV.     From  D,  N,  M,  draw  lines  perpendicular  to  0V. 

The  points  of  intersection  2,  7,  8  are  points  on  the  required  curve.  Proof :  draw 
EC,  FQ,  parallel  to  OV,  and  8  A  parallel  to  OP.  In  the  similar  tri- 
angles O 6 B,  OMA,  we  have  6 B :  MA  ::OB:OA,  or  8  A  :  CB : :  EC :  FS, 
whence  8.4  x  FS  =  CB  x  EC,  or  P8V8  =  PCVC. 

93.   Alternative  Method.     In  Fig.  25  let  b  be  a  known  point  on  the 
curve.     Draw  aD  through  b  and  lay  off  DA  =  ab.     Then  A  is  another 
point  on  the  curve.     Additional  points  may  be  found  by  either  of  the 
constructions    indicated:     e.g.  by  drawing  dh  and   laying  off   hf=  db, 
or  by  drawing  BK  and  laying  off  Kf=  BA.     These  methods  are  prac- 
tically applied  in  the  examination  of  the  expansion  lines  of  steam 
engine  indicator  diagrams. 


\ 


j    K 


94.   Theorem:   Along  an  isothermal  path  for  a  per- 
fect gas,  the  external  work  done  is  equivalent  to 
the  heat  absorbed  (Art.  78). 

internal   energy 


FIG.  25.     Art.  93.  — Second  Method  for  Plotting 
Hyperbolas. 


is  unchanged,  as  indi- 
cated by  Joule's  law 
(Art.  75)  ;  hence  the  expenditure  of  heat  is  solely  for  the  performance 
of  external  work.  H=  T+  1+  W,  but  T  =  0,  T+I=Q,  and  H=  W. 

Conversely,  we  have  Mayer's  principle,  that  "  the  work  done  in  compressing  a 
portion  of  gas  at  constant  temperature  from  one  volume  to  another  is  dynamically 
equivalent  to  the  heat  emitted  by  the  gas  during  the  compression"  (2). 

95.  Work  done  during  Isothermal  Expansion.  To  obtain  the  ex- 
ternal work  done  under  any  portion  of  the  isothermal  curve,  Fig.  24, 
we  must  use  the  integral  form, 

W=$vPdV 


46  APPLIED  THERMODYNAMICS 

in  which  v,  I7' are  the  initial  and  final  volumes.  But,  from  the  equa- 
tion of  the  curve,  pv  =  P  F",  P  —  pv  -r-  F",  and  when  p  and  v  are  given, 
V  V  V  p 

I/         "  1)  1)  _L 

The  heat  absorbed  is  equal  to  this  value  divided  by  778.  For  V  =  infinity,  this 
expression  is  itself  equal  to  infinity ;  the  external  work  area  under  an  indefinitely 
extended  isothermal  is  infinite. 

96.  Perfect  Gas  Isodynamic  (Art.  87).     Since  in  a  perfect  gas  the 
internal  energy  is  fixed  by  the  temperature  alone,  the  internal  energy 
along  an  isothermal  is  constant,  and  the  isodynamic  and  isothermal 
paths  coincide. 

97.  Expansion  in  General.     We   may   for   the  present  limit  the 
consideration  of  possible  paths  to  those  in  which  increases  of  volume 
are  accompanied  by  more  or  less  marked  decreases  in  pressure ;  the 
latter  ranging,  say,  from  zero  to  infinity  in  rate.     If  the  volume  in- 

n=0        creases  without  any  fall  in  pressure,  the 
path  is  one   of  constant  pressure ;  if  the 
volume   increases   only    when  the  fall  of 
pressure  is  infinite,  the  path  is  one  of  con- 
stant volume.     The  paths  under  considera- 
tion will  usually  fall  between  these  two, 
FIG.  26.    Art.  07. —Expansive   like  ak  ac>  a^  etc->  Fig.  26.     The  general 
Paths-  law  for  all  of  these  paths  is  P  Vn  =  a  con- 

stant, in  which  the  slope  is  determined  by  the  value  of  the  exponent  n 
(Art.  91).  For  w  =  0,  the  path  is  one  of  constant  pressure,  ae,  Fig.  26. 
For  n—  infinity,  the  path  is  one  of  constant  volume.  The  " steepness" 
of  the  path  increases  with  the  value  of  n.  (Note  that  the  exponent 
n  applies  to  V  only,  not  to  the  whole  expression.) 

98.  Work  done  by  Expansion.     For  this  general  case,  the  external 
work  area,  adopting  the  notation  of  Art.  95,  is, 


But  since  pv*  —  PFn,  P  =  pvn  •*•  V*\  whence,  wheiijt?  and  v  are  given, 

+  V)  = 


THE  ADIABATIC  47 

When  V  =  infinity,  P  =  0,  and  the  work  is  indeterminate  by  this  expression  ;  but 

we  may  write  W  =  -^-  (l  -  —  }  =  -^—  fl  -  f-V"1],  ™  which,  for  F=  in- 
n—I \          pv  I      n  —  1  L         \VI      J 

finity,  W  —  pv  -=-  (n  —  l),a  finite  quantity.  The  work  under  an  exponential  curve 
is  thus  finite  and  commensurable,  no  matter  how  far  the  expansion  be  continued. 
For  n  =  1,  the  work  obvious!}7  becomes  infinite  with  infinite  expansion  (Art.  95). 

99.    Relations  of  Properties.     For  a  perfect  gas,  in  which  £_L  —  7^  we  have 

PVt  =  pvT. 

If  expansion  proceeds  according  to  the  law  PVn  =  pvn,  we  obtain,  dividing  the 
first  of  these  equations  by  the  second, 


V*        vn  T      \V' 

This  result  permits  of  the  computation  of  the  change  in  temperature  following  a 
given  expansion.  We  may  similarly  derive  a  relation  between  temperature  and 
pressure.  Since 

i  I 

pcn  =  PFn,  t'(/?)w  =  F(P)*.     Dividing  the  expression  pvT  =  PVt  by  this,  we  have 

8=1  •  tei  *      /pX^1 

T(p)  n  =«(P)  n  ,  whence  -  =(-  )  n  . 

f     v/v 

By  interpretation  of  these  formulas  of  relation,  we  observe  that  for 
values  of  n  exceeding  unity,  during  expansion  (i.e.  increase  of  volume),  the 
pressure  and  temperature  decrease,  while  external  work  is  done.  The 
gain  or  loss  of  heat  we  cannot  yet  determine.  On  the  other  hand,  during 
compression,  the  volume  decreases,  the  pressure  and  temperature  increase, 
and  work  is  spent  upon  the  gas.  In  the  work  expression  of  Art.  98,  if 
p,  v,  t  are  always  understood  to  denote  the  initial  conditions,  and  P,  F,  T, 
the  final  conditions,  then  the  work  quantity  for  a  compression  is  negative. 

100.  Adiabatic  Process.  This  term  (Art.  83)  is  applied  to  any 
process  conducted  without  the  reception  or  rejection  of  heat  from  or 
to  surrounding  bodies  by  the  substance  under  consideration.  It  is 
by  far  the  most  important  mode  of  expansion  which  we  shall  have  to 
consider.  The  substance  expands  without  giving  heat  to,  or  taking 
heat  from,  other  bodies.  It  may  lose  heat,  by  doing  work;  or,  in  com- 
pression, work  may  be  expended  on  the  substance  so  as  to  cause  it  to 
gain  heat  :  but  there  is  no  transfer  of  heat  between  it  and  surrounding 
bodies.  If  air  could  be  worked  in  a  perfectly  non-conducting  cylinder, 
we  should  have  a  practical  instance  of  adiabatic  expansion.  In 
practice  we  sometimes  approach  the  adiabatic  path  closely,  by  causing 
expansion  to  take  place  with  great  rapidity,  so  that  there  is  no  time 


48  APPLIED  THERMODYNAMICS 

for  the  transfer  of  heat.  The  expansions  and  compressions  of  the  air 
which  occur  in  sound  waves  are  adiabatic,  on  account  of  their  rapidity 
(Art.  105).  In  the  fundamental  equation  ff=  T  +  1+  W,  the  adi- 
abatic process  makes  H—  0,  whence  W—  —(T+I)\  or,  the  external 
work  done  is  equivalent  to  the  loss  of  internal  energy^  at  the  expense  of 
which  energy  the  work  is  performed. 

101.  Adiabatic  Equation.  Let  unit  quantity  of  gas  expand  adiabatically 
to  an  infinitesimal  extent,  increasing  its  volume  by  dv,  and  decreasing  its 
pressure  and  temperature  by  dp  and  dt.  As  has  just  been  shown, 
W=  —  (T  +  /),  the  expression  in  the  parenthesis  denoting  the  change  in 
internal  energy  during  expansion.  The  heat  necessary  to  produce  this 
change  would  be  Idt,  I  being  the  specific  heat  at  constant  volume.  The  ex- 
ternal work  done  is  W=  pdv  ;  consequently,  pdv  =  —  Idt.  From  the 

equation  of  the  gas,  pv  =  Hi,  £  =*—  ,  whence,  dt  =  —  (pdv  -f-  vdp).  Using 
this  value  for  dt,  R  R 

pdv  =  --  (pdv  +  vdp). 
R 

But  R  is  equal  to  the  difference  of  the  specific  heats,  or  to  k  —  l\  so  that 

pdv  =  ~  ~ 


ypdv  —  pdv  =  —  pdv  —  vdp, 


y~^  =  —  -P,  giving  by  integration, 
v  p 


ylogev  -}-  Iogej9  =  constant, 
or    pvv  =  constant, 

y  being  the  ratio  of  the  specific  heats  at  constant  pressure  and  con- 
stant volume  (Art.  69.) 

102.  Second  Derivation.  A  simpler,  though  less  satisfactory,  mode  of 
derivation  of  the  adiabatic  equation  is  adopted  by  some  writers.  Assum- 
ing that  the  adiabatic  is  a  special  case  of  expansion  according  to  the  law 
pvn  =  PV",  the  external  work  done,  according  to  Art.  98,  is 

R(t  -  T) 
n-l 


ADIABATIC   EXPANSION  49 

During  a  change  of  temperature  from  t  to  T,  the  change  in  internal  energy 
is  l(t  —  T),  or  from  Art.  70,  since  I  =  It  -*-(y  —  1),  it  is 

R(t  -  T) 

y-i 

But  in  adiabatic  expansion,  the  external  icork  done  is  equivalent  to  the 
change  in  internal  energy  ;  consequently 

R(t-  T)  =  R(t-  T\ 
n—  1  y—  1 

n  =  y,  and  the  adiabatic  equation  isjwf  =  PF*.     For  air,  the  adiabatic  is 
then  represented  by  the  expression  p(v)lm¥e  =  a  constant. 

103.  Graphical  Presentation.      Since  along  an  adiabatic  the   external 
work  is  done  at  the  expense  of  the  internal  energy,  the  temperature  must 
fall  during  expansion.     In  the  diagram  of  Fig.  19,  this  is  shown  by  com- 
paring the  line  ab,  an  isothermal,  with  ae,  an  adiabatic.     The  relation  of 
p  to  v,  in  adiabatic  expansion,  is  such  as  to  cause  the  temperature  to  fall. 
The   projections    of    these   two   paths   on   the  pv  plane  show  that    as 
expansion    proceeds   from    a,    the    pressure    falls    more    rapidly  along 
the  adiabatic  than  along  the  isothermal,  a  result  which  might  have  been 
anticipated  from  comparison  of  the  equations  of  the  two  paths.     If  an 
isothermal  and  an  adiabatic  be  drawn  through  the  same  point,  the  latter 
will  be  the  "  steeper  "  of  the  two  curves.     Any  number  of  adiabatics  may 
be  constructed  on  the  pv  diagram,  depending  upon  the  value  assigned  to 
the  constant  (pv*)  ;  but  since  this  value  is  determined,  for  any  particular 
perfect  gas,  by  contemporaneous  values  of  p  and  v,  only  one  adiabatic  can 
be  drawn  for  a  given  gas  through  a  given  point. 

104.  Relations  of  Properties.     By  the  methods  of  Art.  98  and 
Art.  99,  we  find,  for  adiabatic  changes, 


During  expansion,  the  pressure  and  temperature  decrease,  external  work  is  done 
at  the  expense  of  the  internal  energy,  and  there  is  no  reception  or  rejection  of  heat. 

105.    Direct  Calculation  of  the  Value  of  y-     The  velocity  of  a  wave  in  an 

elastic  medium  is,  according  to  a  fundamental  proposition  in  dynamics,  directly 
as  the  square  root  of  the  coefficient  of  elasticity  divided  by  the  density  of  the 
medium  ;  or,  for  ultimate  values, 

v  =  Ve  -T-  d. 

Let  g  denote  the  acceleration  due  to  gravity,  w  the  weight  of  unit  volume  of  the 
medium  at  the  density  d,  m  the  weight  of  unit  volume  of  mercury,  and  b  the 


50 


APPLIED  THERMODYNAMICS 


height  of  the  mercurial  barometer.  For  unconfined  air  at  constant  pressure, 
the  pressure  equals  the  elasticity  ;  for  Idp  =  —  eds,  in  which  dp  is  an  infinitesimal 
increment  of  pressure  applied  to  a  body  of  length  Z,  producing  an  extension  ds, 
equivalent  to  a  compression  —  ds;  and  if  the  body  be  a  gas  kept  at  constant  tem- 
perature, pv  —  e,  pdv  =  -  vdp  ;  and  if  its  form  be  prismatic  and  its  cross  section 
unity,  such  that  /  =  r,  then  dv  =  ds,  pds  =  —  Idp,  and  p  =  —  Idp  -+-  ds  =  e.  Then 
e  =p  =  bm,  and  since  d  =  w  -4-  g,  we  have 

v  =  ^bmg  H-  w. 

This  would  be  the  velocity  of  sound  in  air,  for  example,  if  there  were  no  change 
in  temperature.  But  the  vibrations  which  constitute  sound  are  accompanied  by 
changes  in  temperature;  these  changes  are  adiabatic  (Art.  100),  and  it  has  been 
shown  (Art.  104),  that  the  pressure  varies  during  such  changes  inversely  as  that 
power  of  the  specific  v'olume  whose  exponent  is  y  ;  or  directly  as  the  y  power  of 
the  density.  Then  e  =  (f)dy.  Taking  the  expression  first  given,  and  putting  in 

differential  form, 

e 


But   if   e  =  (/Xy,  we  have,  say,  e  =  ad*,  de  =  yady~ldd,  a  =  e  -f-  d»,  de  =  y-  •  dd 


At  the  temperature  of  melting  ice,  when  b  =  2.494  ft.,  v  has  been  found  by  experi- 
ment to  be  1089  ft.  per  second ;  whence 

_  wv2  =  0.081  x  1089  x  1089  _  1  4n  g. 

y~bmg~  32.19 x 2.494  x  849.3  ~ 

106.  Representation  of  Heat  Absorbed.  Theorem :  The  heat  ab- 
sorbed on  any  path  is  represented  on  the  PV  diagram  by  the  area  en- 
closed between  that  path  and  the  two  adiabatics  through  its  extremities, 
indefinitely  prolonged  to  the  right. 

Let  the  path  be  ab,  Fig.  27.     Draw  the  adiabatics  an,  bN.     These 
may  be  conceived  to  meet  at  an  infinite  dis- 
tance to  the  right,  forming  with  the  path  the 
closed  cycle    abNn.      In  such  closed   cycle, 
the  total  expenditure  of  heat  is,  from  Art. 
90,  represented  by   the  enclosed  area ;    but 
since  no  heat  is  absorbed  or   emitted  along 
FIG.  27.  Arts.  106,109.— Rep-  the  adiabatics,  all  of  the  heat  changes  in  the 
resentation    of    Heat    Ab-  CyC^e  must  have  occurred  along"  the  path  ab* 

sorbed. 

and  this  change  of  heat  is  represented  by  the 

area  abNn.     If  the  path  be  taken  in  the  reverse  direction,  i.e.  from  b 
to  a,  the  area  abNn  measures  the  heat  emitted. 


GRAPHICAL  REPRESENTATIONS 


51 


107.  Representations  of  Thermal  Capacities.     Let  ab,  cd,  Fig.  28,  be  two 
isothermals,  differing  by  one  degree.     Then  efnN  represents  the  specific 
heat  cut  constant  volume,  egmN  the  specific  heat  at 

constant  pressure,  eN,  fn,  and  gm  being  adiabatics. 
The  latter  is  apparently  the  greater,  as  it  should 
be.  Similarly,  if  ab  denotes  unit  increase  of 
volume,  the  area  abMX  represents  the  latent  heat 
of  expansion.  The  other  thermal  capacities  men- 
tioned in  Art.  58  may  be  similarly  represented. 

FIG.  28.    Art.  107. — Thermal 
Capacities 

108.  Isodiabatics.    Let  AD,  Fig,  29,  represent 

any  path  following  the  law  pv*  =  PV*,  intersecting  the  two  isothermals 

CX,  B  Y.  The  heat  absorbed  along  this 
path  may  be  represented  by  the  area 
nADN,  An  and  DN  being  adiabatics.  If 
some  other  path,  BC,  be -found,  in  which 
pv*  =  PV*t  the  value  of  n  being  the 
same  as  that  for  the  path  AD,  this  path 
connecting  the  same  two  isothermals, 
then  the  two  paths  AD  and  BC  are 
called  isodiabatics,  and  as  will  appear 
(Art.  112),  the  areas  mCBM  and  nADN 
are  equal. 

Theorem.    The  ratios  of  pressures  or  of  volumes  at  points  on  isodiabatics 
intersected  by  isothermals  are  constant. 

In  Fig.  29.  if  we  designate  the  pressures  at  A,  D,  C,  and  B  by  PA,  PD, 


FIG.  29.    Art.  108.  — Isodiabatics. 


Pc,  PB,  respectively,  then  from  Art.  99, 


PB 


ri  =  Pj> 
PA 


So  also, 


-=&}      H^C)      ,  whence -^  =  ^ 


t 


VA 


109.  Derivation  of  Joule's  Law.  From  the  theorem  of  Art.  106,  Rankine 
has  established  in  a  very  simple  manner  the  principle  of  Joule,  that  the 
change  of  internal  energy  along  any  jxith  of  any  substance  depends  upon  the 
initial  and  final  states  alone,  and  not  upon  the  nature  of  the  path.  In 
Fig.  27,  draw  the  vertical  lines  ax,  by.  The  total  heat  absorbed  along 
ab  —  nabN,  the  external  work  done  =  xaby.  The  difference  =  nabN—xaby 
=  nzbN—  xazy,  is  the  change  in  internal  energy  ;  H=  T  +  /+  W,  whence 
H—  W  =  (T+  /);  and  the  extent  of  these  areas  is  unaffected  by  any 
change  in  the  path  ab,  so  long  as  the  points  a  and  b  remain  fixed. 


52  APPLIED  THERMODYNAMICS 

This  demonstration  is  of  major  importance  because  it  establishes 
the  cardinal  nature  of  internal  energy  for  all  substances  in  uniform 
thermal  condition.  Compare  Art.  90,  footnote. 

110.  Value  of  y.  A  method  of  computing  the  value  of  y  for  air  has 
been  given  in  Art.  105.  The  apparatus  shown,  in  Fig.  30  has  been  used 
by  several  observers  to  obtain  direct  values  for  various  gases.  The  vessel 
was  filled  with  gas  at  P,  V,  and  T,  T  being  the  temperature  of  the  atmos- 
phere,  and  P  a  pressure  somewhat  in  excess  of  that 

/• flQ^  of  the  atmosphere.     By   opening  the   stopcock,  a 

sudden  expansion  took  place,  the  pressure  falling 
to  that  of  the  atmosphere,  and  the  temperature 
falling  to  a  point  considerably  below  that  of  the 
atmosphere.  Let  the  state  of  the  gas  after  this 
adiabatic  expansion  be  p,  v,  t.  Then,  since 


FIG.  30.      Art.  110.  -De-  y  =  logff-logP 

sormes'  Apparatus.  log  V  —  log  V 

After  this  operation,  the  stopcock  is  closed,  and  the  gas  remaining  in  the 
vessel  is  allowed  to  return  to  its  initial  condition  of  temperature,  T. 
During  this  operation,  the  volume  remains  constant;  so  that  the  final 
state  isp2>  v>  T;  whence  p2v  =  PF,  or  log  V  '  —  log  v  =  Iogp2  —  log  P.  Sub- 
stituting this  value  of  log  V  —  log  v  in  the  expression  for  y,  we  have 


^ 
logp2-logP' 

so  that  the  value  of  y  may  be  computed/rom  the  pressure  changes  alone. 
Clement  and  Desormes  obtained  in  this  manner  for  air,  y  =  1.3524;  Gay- 
Lussac  and  Wilter  found  y  =  1.3745.  The  experiments  of  Him,  Weisbach, 
Masson,  Cazin,  and  Kohlrausch  were  conducted  in  the  same  manner.  The 
method  is  not  sufficiently  exact. 

111.   Expansions  in  General.     In  adiabatic  expansion,  the  external  work 
done  and  the  change  in  internal  energy  are  equally  represented  by  the 

expression  ^-  ~—  —  ,  derived  as  in  Art.  98.     For  expansion  from  p,  v  to 

pv 
infinite  volume,  this  becomes     _-.  •     The  external  work  done  during  any 

7~>T7" 

expansion  according  to  the  law  pvn  =  PVn  from  pv  to  PF,  is  W=^v~~        • 


The  stock  of  internal  energy  at  p,  v,  is  -—  =  It:  at  P,  F,  it  is        -  =  IT. 

y—  i  y-i 

The  total  heat  expended  during  expansion  is  equal  to  the  algebraic  sum 
of    the   external   work    done   and  the    internal   energy   gained.     Then, 


POLYTROPIC  PATHS  53 

=(          pr/    1  1    N 

\n-l      y-lj 


=  l(t—  T)(y~™\  in  which  £  is  the  initial,  and  T  the  final  temperature. 


This  gives  a  measure  of  the  net  heat  absorbed  or  emitted  during  any  ex- 
pansion or  compression  according  to  the  law  pvn  =  constant.  When  n 
exceeds  y,  the  sign  of  H  is  minus  ;  heat  is  emitted  ;  when  n  is  less  than  y 
but  greater  than  1.0,  heat  is  absorbed  :  the  temperature  falling  in  both 
cases.  When  n  =  y,  the  path  is  adiabatic,  and  heat  is  neither  absorbed 
nor  emitted. 

112.  Specific  Heat.     Since  for  any  change  of  temperature  involving 
a  heat  absorption  H,  the  mean  specific  heat  is 

s=r^? 

we  derive  from  the  last  equation  of  Art.  Ill  the  expression, 

_  7  n  -  y 
s  —  i  —  —  , 

71  —  1 

giving  the  specific  heat  along  any  path  pvn  =  PVn.  Since  the  values 
of  n  are  the  same  for  isodiabatics,  the  specific  heats  along  such  paths  are 
equal  (Art.  108). 

113.  Ratio  of  Internal  Energy  Change  to  External  Work.      For  any    given 
value  of  n,  this  ratio  has  the  constant  value 

n-1 


114.  Poly  tropic  Paths.  A  name  is  needed  for  that  class  of  paths 
following  the  general  law  pvn  =  P  Vn,  a  constant.  Since  for  any 
gas  y  and  I  are  constant,  and  since  for  any  particular  one  of  these 
paths  n  is  constant,  the  final  formula  of  Art.  Ill  reduces  to 


In  other  words,  the  rate  of  heat  absorption  or  emission  is  directly  pro- 
portional to  the  temperature  change  ;  the  specific  heat  is  constant.  Such 
paths  are  called  polytropic.  A  large  proportion  of  the  paths  exempli- 
fied in  engineering  problems  may  be  treated  as  polytropics. 


54 


APPLIED  THERMODYNAMICS 


115.  Relations  of  n  and  s.  We  have  discussed  such  paths  in  which  the 
value  of  n  ranges  from  1.0  to  infinity.  Figure  31  will  make  the  concep- 
tion more  general.  Let  a  represent  the  initial  condition  of  the  gas.  If 

p 

g 


p 


FIG.  31.    Art.  115.  —  Polytropic  Paths. 

it  expands  along  the  isothermal  ab,  n  =  1,  and  s,  the  specific  heat,  is  infi- 
nite ;  no  addition  of  heat  whatever  can  change  the  temperature.  If  it 
expands  at  constant  pressure,  along  ae,  n  =  0,  and  the  specific  heat  is  finite 
and  equal  to  ly  =  ~k.  If  the  path  is  ag,  at  constant  volume,  n  is  infinite 
and  the  specific  heat  is  positive,  finite,  and  equal  to  I.  Along  the  isother- 
mal of  (compression),  the  value  of  n  is  1,  and  s  is  again  infinite.  Along 
the  adiabatic  oh,  n  =  1.402  and  s  =  0.  Along  ai,  n  =  0  and  s  =  k.  Along 
ad,  n  is  infinite  and  s  =  /.  Most  of  these  relations  are  directly  derived 
from  Art.  112,  or  may  in  some  cases  be  even  more  readily  apprehended  by 
drawing  the  adiabatics,  en,  gN,  fm,  iM,  dp,  bP,  and  noting  the  signs  of  the 
areas  representing  heats  absorbed  or  emitted  with  changes  in  temperature. 
For  any  path  lying  between  ah  and  af  or  between  ac  and  ab,  the  specific 
heat  is  negative,  i.e.  the  addition  of  heat  cannot  keep  the  temperature  from  fall- 
ing :  nor  its  abstraction  from  rising. 


116.    Relations  of  Curves  :   Graphical  Representation  of  n.     Any  number  of 
curves  may  be  drawn,  following  the  law  pvn  —  C,  as  the  value  of  C  is  changed. 


RELATIONS   OF  n  AND  s 


55 


In  Fig.  32,  let  ab,  cd,  <?/be  curves  thus  drawn.     Their  general  equation  is  pv*  =  C, 
whence 

n 

W*.  \o/ 

dv         v 

If  MTV  is  the  angle  made  by 
the  tangent  to  one  of  the  curves 
with  the  axis  OF,  and  A/OF 
the  angle  formed  by  the  radius 
vector  R  M  with  the  axis  OF, 
then,  since  dp  -H  dv  is  the  tan- 
gent of  MTV,  and p  -f-  v  is  the 
tangent  of  3/0  F, 


FIG.  32.    Art.  116. — Determination  of  Exponent. 
-  tan  MTV  =  n  tan  MO  V. 


If  the  radius  vector  be  produced  as  RMNQ,  the  relations  of  the  angles  made  be- 
tween the  OF  axis  and  the  successive  tangents  MT,  NS,  QU,  are  to  the  angle 
P  J/OFas  just  given;  hence  the  various  tangents 

are  parallel  (4). 

/  Since  tan  MTV  =  Mg  +  g T  and  tan  M 0 V  = 

Mg  -T-  Og,  the  preceding  equation  gives 


9T          Ogf 

whence  n  =  Og  -r-  gT.  (The  algebraic  signs  of 
Og  and  gT,  measured  from  g,  are  different.)  In 
order  to  determine  the  value  of  n  from  a  given 
curve,  we  need  therefore  only  draw  a  tangent 
MT  and  a  radius  vector  MO,  whence  by  drop- 
ping the  perpendicular  Mg  the  relation  Og  4-  gT 
is  established.  If  we  lay  off  from  0  the  distance 
OA  as  a  unit  of  length,  drawing  A  C  parallel  to 
the  tangent,  and  CB  through  C,  parallel  to  the 


/     /B 

T 


FIG.  33.    Art.  116.  — Negative 
Exponent. 


radius  vector,  then  by  similar  triangles 
Og  :  gT: :  OB  :  OA  and  Og  -  gT  =  OB  =  n. 
Figure  33  illustrates  the  generality  of  this 
method  by  showing  its  application  to  a 
curve  in  which  the  value  of  n  is  negative. 

117.  Plotting  of  Curves:  Brauer's 
Method.  The  following  is  a  simple  method 
for  the  plotting  of  exponential  curves,  in- 
cluding the  adiabatic,  which  is  ordinarily 
a  tedious  process.  Let  the  point  M, 
Fig.  34,  be  given  as  one  point  on  the  re- 
quired curve.  Draw  a  line  OA  making  an 
angle  F0.4  with  the  axis  OF,  and  a  line 
OB  making  an  angle  POB  with  the  axis 


Fia.  34.    Art.  117.  — Brauer's  Method! 


56  APPLIED  THERMODYNAMICS 

OP.  Draw  the  vertical  line  MS  and  the  horizontal  line  MT.  Also  draw  the 
line  TU  making  an  angle  of  45°  with  OP,  and  the  line  SR  making  an  angle  of 
45°  with  MS.  Draw  the  vertical  line  RN  through  R,  and  the  horizontal  line  UN 
through  U.  The  coordinates  of  the  point  of  intersection,  N,  of  these  lines,  are 
OR  and  RN.  Let  the  coordinates  of  M,  TM  (=  OQ),  and  MQ  be  designated  by 
v,  p  ;  and  those  of  N,  OR,  and  RN  (=OL),  by  V,  P.  Then  tan  VOA  =  QS  -=-  OQ 
=  QR  +  TM  =  (V-v)  +  v;  and  tan  POB=  UL  H-  OL=  TL  +  NR  =  (p  -  P)  -P; 
whence  F  =  v  (tan  VOA  +  1)  and  p  =  P  (tan  POB  +  1).  If  the  law  of  the 
curve  through  M  and  N  is  to  be  pvn  =  P  Vn,  we  obtain 

P(tanPOB  +  !)?»»=  P{u(tan  VOA  +  l)}n, 

whence  (tan  POB  +  1)  =  (tan  FOJ  +  l)n.  If  now,  in  the  first  place,  we  make  the 
angles  POB,  VOA  such  as  to  fulfill  this  condition,  then  the  point  N  and  others 
similarly  determined  will  be  points  on  a  curve  following  the  Iawj9yn  =  PFn. 

118.  Tabular  Method.     The  equation  pvn  =  PVn  may  be  written  p  =  p(Z.\* 

or  log  jo  —  log  P  =  n  log  (  V  •*•  v).  If  we  express  P  as  a  definite  initial  pressure  for 
all  PVn  curves,  then  for  a  specific  value  of  n  and  for  definite  ratios  V  •*•  v  we  may 
tabulate  successive  values  of  logjt?  and  of  p.  Such  tables  for  various  values  of  n 
are  commonly  used.  In  employing  them,  the  final  pressure  is  found  in  terms  of 
the  initial  pressure  for  various  ratios  of  final  to  initial  volume. 

119.  Representation  of  Internal  Energy.     In  Fig.  35,  let  An  represent 
an  adiabatic.     During  expansion  from  A  to  a,  the  external  work  done  is 

Aabc,  which,  from  the  law  of  the  adiabatic,  is 
equal  to  the  expenditure  of  internal  energy.  If 
expansion  is  continued  indefinitely,  the  adiabatic 
An  gradually  approaches  the  axis  OF,  the  area 
below  it  continually  representing  expenditure  of 
internal  energy,  until  with  infinite  expansion  An 
and  OF  coincide.  The  internal  energy  is  then  ex- 

FIG.  35.  Art.  119.—  Repre-  nausted.  The  total  internal  energy  of  a  substance 
sentation  of  Internal  may  therefore  be  represented  by  the  area  between 
Energy-  the  adiabatic  through  its  state,  indefinitely  prolonged 

to  the  right,  and  the  horizontal  axis.    Kepresenting  this  quantity  by  E,  then 

from  Art.  Ill, 


y  —  1 

where  v  is  the  initial  volume,  p  the  initial  pressure,  and  y  the  adiabatic 
exponent.     This  is  a  finite  and  commensurable  quantity. 

120.    Representation  by  Isodynamic  Lines.     A  defect  of  the  preceding 
representation  is  that  the  areas  cannot  be  included  on  a  finite  diagram. 


GRAPHICAL  REPRESENTATIONS 


57 


In  Fig.  36,  consider  the  path  AB.  Let  BC  be  an  adiabatic  and  AC  an- 
isodynamic.  It  is  required  to  find  the  change  of  internal  energy  between 
A  and  B.  The  external  work  done  daring  adi- 
abatic expansion  from  B  to  C  is  equal  to  BCcb ; 
and  this  is  equal  to  the  change  of  internal  en- 
ergy between  B  and  C.  But  the  internal  energy 
is  the  same  at  C  as  at  A,  because  AC  is  an 
isodynamic.  Consequently,  the  change  of  in- 
ternal energy  between  A  and  B  is  represented 
by  the  area  BCcb;  or,  generally,  by  the  area 
included  between  the  adiabatic  through  the  final 
state,  extended  to  its  intersection  with  the  iso- 
dynamic through  the  initial  state,  and  the  hori- 
zontal axis. 


FIG.  36.  Arts.  120,  121.  — In- 
ternal Energy,  Second  Dia- 
gram. 


FIG.  37.     Art.  121.  —External 
Work  and  Internal  Energy. 


121.  Source  of  External  Work.     If  in  Fig.  36  the  path  is  such  as  to  increase 

the  temperature  of  the  substance,  or  even  to  keep  its 
e  temperature  from  decreasing  as  much  as  it  would 

along  an  adiabatic,  then  heat  must  be  absorbed. 

\  d  ^^  Thus,  comparing  the  paths  ad  and  ac,  Fig.  37,  aN 

and  cm  being  adiabatics,  the  external  work  done 
along  ad  is  adef,  no  heat  is  absorbed,  and  the  internal 
energy  decreases  by  adef.  Along  ac,  the  external 
work  done  is  acef,  of  which  a^/e/was  done  at  the  ex- 
pense of  the  internal  energy,  and  acd  by  reason  of 
the  heat  absorbed.  The  total  heat  absorbed  was 

Nacm,  of  which  acd  was  expended  in  doing  external  work,  while  Ndcm  went 

to  increase  the  stock  of  internal  energy. 

122.  Application  to  Isothermal  Expansion.     If  the  path  is  isothermal,  Fig.  38, 
line  AB,  then  if  BN,  An  are  adiabatics,  we  have, 

W  +  X  =  external  work  done, 

X  +  Y  =  heat  absorbed  =  W  +  X, 

W  +  Z  =  internal  energy  at  A, 

Y  +  Z  =  internal  energy  at  B, 

W  =  work  done  at  the  expense  of  the  in- 
ternal energy  present  at  .1, 

X  =  work  done  by  reason  of  the  absorption 
of  heat  along  AB, 

Z  =  residual  internal  energy  of  that  originally 
present  at  A, 

Y  =  additional  internal  energy  imparted  by 
the  heat  absorbed ; 
and  since  in  a  perfect  gas  isothermals  are  isodynamics,  we  note  that 

W  +  Z  =  Y  +  Z  and  W  =  F(5). 


FIG.  38.      Art.   122.  — Heat    and 
Work  in  Isothermal  Expansion. 


58 


APPLIED  THERMODYNAMICS 


123.    Finite  Area  representing  Heat  Expenditure.     In  Fig.  39,  let  ab  be  any 
path,  bn  and  aN  adiabatics,  and  ac  an  isodynamic.     The  external  work  done  along 

ab  is  abde ;  while  the  increase  of  internal  energy  is 
bcfd.  The  total  heat  absorbed  is  then  represented  by 
the  combined  areas  abcfe.  If  the  path  ab  is  iso- 
thermal, this  construction  leads  to  the  known  result 
that  there  is  no  gain  of  internal  energy,  and  that  the 
total  heat  absorbed  equals  the  external  work.  If  the 
path  be  one  of  those  de- 
scribed in  Art.  115  as  of 
negative  specific  heat,  we 
may  represent  it  as  a</, 
Fig.  40.  Let  bgm  be  an 
adiabatic.  The  external 

work  done  is  agde.  The  change  of  internal  energy, 
from  Art.  120,  is  bydf,  if  ab  is  an  isodynamic;  and 
this  being  a  negative  area,  we  note  that  internal  en- 
ergy has  been  expended,  although  heat  has  been  ab- 
sorbed. Consequently,  the  temperature  has  fallen.  It 
seems  absurd  to  conceive  of  a  substance  as  receiving  heat  while  falling  in  tem- 
perature. The  explanation  is  that  it  is  cooling,  by  doing  external  work,  faster 
than  the  supply  of  heat  can  warm  it.  Thus,  H  =  T  +  I  +  IF;  but  H  <  W]  con- 
sequently, ( T  +  /)  is  negative. 


FIG.  39.     Art.  123.— Represen- 
tation of  Heat  Absorbed. 


FIG.  40.     Art.  123.  — Nega- 
tive Specific  Heat. 


MODIFICATIONS  IN  IRREVERSIBLE  PROCESSES 

124.  Constrained  and  Free  Expansion.     In  Art.  86  it  was  assumed  that 
the  path  of  the  substance  was  one  involving  changes  of  volume  against  a 
resistance.     Such  changes  constitute  constrained  expansion.     In  this  pre- 
liminary analysis,  they  are  assumed  to   take  place   slowly,  so   that   no 
mechanical  work  is  done  by  reason  of  the  velocity  with  which  they  are 
effected.     When  a  substance  expands  against  no  resistance,  as  in  Joule's 
experiment,  or  against  a  comparatively  slight  resistance,  we  have  what  is 
known  as  free  expansion,  and  the  external  work  is  wholly  or  partly  due 
to  velocity  changes. 

125.  Reversibility.     All  of  the  polytropic  curves  which  have  thus  far 
been  discussed  exemplify  constrained  expansion.     The  external  and  in- 
ternal pressures  at  any  state,  as  in   Art.  86,  differ  to  an  infinitesimal 
extent  only ;  the  quantities  are  therefore  in  finite  terms  equal,  and  the 
processes  may  be  worked  at  ivill  in  either  direction.     A  polytropic  path 
having  a  finite  exponent  is  in  general,  then,  reversible,  a  characteristic  of 
fundamental  importance.     During  the  adiabatic  process  which  occurred 
in  Joule's  experiment,  the  externally  resisting  pressure  was  zero  while 
the  internal  pressure  of  the  gas  was  finite.     The  process  could  not  be 


IRREVERSIBLE  PROCESSES  59 

reversed,  for  it  would  be  impossible  for  the  gas  to  flow  against  a  pressure 
greater  than  its  own.  The  generation  of  heat  by  friction,  the  absorption 
of  heat  by  one  body  from  another,  etc.*  are  more  familiar  instances  of 
irreversible  process.  Since  these  actions  take  place  to  a  greater  or  less 
extent  in  all  actual  thermal  phenomena,  it  is  impossible  for  any  actual 
process  to  be  perfectly  reversible.  "A  process  aifecting  two  substances  is 
reversible  only  when  the  conditions  existing  at  the  commencement  of  the 
process  may  be  directly  restored  without  compensating  changes  in  other 
bodies." 

126.  Irreversible  Expansion.  In  Fig.  41,  let  the  substance  expand 
unconstrainedly,  as  in  Joule's  experiment,  from  a  to  6,  this  expansion 
being  produced  by  the  sudden  decrease  in  ex- 
ternal pressure  when  the  stopcock  is  opened. 
Along  the  path  a&,  there  is  a  violent  movement  of 
the  particles  of  gas;  the  kinetic  energy  thus 
evolved  is  transformed  into  pressure  at  the  end 
of  the  expansion,  causing  a  rise  of  pressure  to  c. 
The  gain  or  loss  of  internal  energy  depends  solely 


upon  the  states  a,  c;  the  external  work  done  does      FIG.  41.    Art.  126.—  irre- 
not  depend  on  the  irreversible  path  ab,  for  with  versible  Path. 

a  zero  resisting  pressure  no  external  work  is  done.  The  theorem  of  Art.  86 
is  true  only  for  reversible  operations. 

127.  Irreversible  Adiabatic  Process.  Careful  consideration  should  be 
given  to  unconstrained  adiabatic  processes  like  those  exemplified  in  Joule's 
experiment.  In  that  instance,  the  temperature  of  the  gas  was  kept  up  by 
the  transformation  back  to  heat  of  the  velocity  energy  of  the  rapidly 
moving  particles,  through  the  medium'  of  friction.  We  have  here  a  special 
case  of  heat  absorption.  No  heat  was  received  from  tvithout  ;  the  gas 
remained  in  a  heat-insulated  condition.  While  the  process  conforms  to 
the  adiabatic  definition  (Art.  83),  it  involves  an  action  not  contemplated 
when  that  definition  was  framed,  viz.,  a  reception  of  heat,  not  from  sur- 
rounding bodies,  but  from  the  mechanical  action  of  the  substance  itself.  The 
fundamental  formula  of  Art.  12  thus  becomes 


in  which  F'may  denote  a  mechanical  effect  due  to  the  velocity  of  the  parti- 
cles of  the  substance.  This  subject  will  be  encountered  later  in  important 
applications  (Arts.  175,  176,  426,  513). 

(1)  Thermodynamics,  1907,  p.  18.  (2)  Alexander,  Treatise  on  Thermodynamics, 
1803,  p.  105.  (3)  Wormell,  Thermodynamics,  123  ;  Alexander,  Thermodynamics, 
103  ;  Rankine,  The  Steam  Engine,  249,  321  ;  Wood,  Thermodynamics,  71-77,  437. 
(4)  Zeuner,  Technical  Thermodynamics,  Klein  tr.,  1,  156.  (5)  Ripper,  Steam  Engine 
Theory  and  Practice,  1895,  17. 


60  APPLIED  THERMODYNAMICS 


SYNOPSIS   OF   CHAPTER  V 

Pressure,  volume  and  temperature  as  thermodynamic  coordinates. 

Thermal  line,  the  locus  of  a  series  of  successive  states  ;  path,  a  projected  thermal  line. 

Paths :    isothermal,    constant  temperature ;    isodynamic,   constant    internal   energy ; 

adiabatic,  no  transfer  of  heat  to  or  from  surrounding  bodies. 
The  geometrical  representation  of  the  characteristic  equation  is  a  surface. 
The  P  V  diagram :   subtended  areas  represent  external  work ;  a  cycle  is  an  enclosed 

figure  ;  its  area  represents  external  work  ;  it  represents  also  the  net  expenditure  of 

heat. 
The  isothermal :  pvn  =  c,  in  which  n  =  1,  an  equilateral  hyperbola ;  the  external  work 

done  is  equivalent  to  the  heat  absorbed,  =  pv  \oge  —  :  with  a  perfect  gas,  it  coin- 
cides with  the  isodynamic. 

Paths  in  general :  pvn  =  c  ;  external  work  =PV~PV.  L  =  ( ^-V~";  ^=  (*  f*\ 

n  —  l       1      \V  I          t      \pj 

The  adiabatic :  the  external  work  done  is  equivalent  to  the  expenditure  of  internal 
energy ;  pvu  =  c  ;  y  =  1 .402  ;  computation  from  the  velocity  of  sound  in  air. 

The  heat  absorbed  along  any  path  is  represented  by  the  area  between  that  path  and 
the  two  projected  adiabatics  ;  representation  of  k  and  I. 

Isodidbatics.:  n\  =  n%  ;  equal  specific  heats. 

Rankine's  derivation  of  Joule's  law:  the  change  of  internal  energy  between  two  states 
is  independent  of  the  path. 

Apparatus  for  determining  the  value  of  y  from  pressure  changes  alone. 

Along  any  path  pvn  =  c,  the  heat  absorbed  is  l(t  —  ^)(*4lf )  5  the  mean  specific  heat 
is  ln~  y .  Such  paths  are  called  polytropics.  Values  of  n  and  s  for  various  paths. 

Graphical  method  for  determining  the  value  of  n  ;  Brauer's  method  for  plotting  poly- 
tropics  ;  the  tabular  method. 

Graphical  representations  of  internal  energy  ;  representations  of  the  sources  of  external 
work  and  of  the  effects  of  heat ;  finite  area  representing  heat  expenditure. 

Irreversible  processes :  constrained  and  free  expansion  ;  reversibility  ;  no  actual  pro- 
cess is  reversible  ;  example  of  irreversible  process ;  subtended  areas  do  not  repre- 
sent external  work  ;  in  adiabatic  action,  heat  may  be  received  from  the  mechanical 
behavior  of  the  substance  itself;  H=  T  +  I  +  W+  V. 


PROBLEMS 

1.  On  a  perfect  gas  diagram,  the  coordinates  of  whiah  are  internal  energy  and 
volume,  construct  an  isodynamic,  an  isothermal,  and  an  isometric  path  through  ^=2, 
F=2. 

2.  Plot  accurately  the   following:    on  the    TV  diagram,  an   adiabatic  through 
T=270,  F=10;  an  isothermal  through   r=300,   F=20:   on  the   TP  diagram,  an 
adiabatic  through  T=230,  P=5;  an  isothermal  through  T=  190,  P=30.     On  the 
EV diagram,  show  the  shape  of  an  adiabatic  through  E  =  240,  V=  10. 

3.  Show  the  isometric  path  of  a  perfect  gas  on  the  PT  plane  ;  the  isopiestic,  on 
the  VT  plane. 


PROBLEMS  61 

4.  Sketch  the  TV  path  of  wax  from  0°  to  290°  F.,  assuming  the  melting  point  to 
be  90°,  the  boiling  point  290°,  that  wax  expands  in  melting,  and  that  its  maximum 
density  as  liquid  is  at  the  melting  point. 

5.  A  cycle  is  bounded  by  two  isopiestic  paths  through  P=  110,  P  =  100  (pounds 
per  square  foot) ,  and  by  two  isometric  paths  through  V  =  20,  V  —  10  (cubic  feet) . 
Find  the  heat  expended  by  the  working  substance. 

6.  Air  expands  isothermally  at  32°  F.  from  atmospheric  pressure  to  a  pressure  of 
5  Ib.  absolute*  per  square  inch.     Find  its  specific  volume  after  expansion. 

7.  Given  an  isothermal  curve  and  the  OF  axis,  find  graphically  the  OP  axis. 

8.  Prove  the  correctness  of  the  construction  described  in  Art.  93. 

9.  Find  the  heat  absorbed  during  the  expansion  described  in  Problem  6. 

10.  Find  the  specific  heat  for  the  path  PFli2  =  c,  for  air  and  for  hydrogen. 

11.  Along  the  path  PF1-2  =  c,  find  the  external  work  done  in  expanding  from 
P  =  1000,  V  =  10,  to  V  —  100.     Find  also  the  heat  absorbed,  and  the  loss  of  internal 
energy,  if  the  substance  is  one  pound  of  air.     Units  are  pounds  per  square  foot  and 
cubic  feet. 

12.  A  perfect  gas  is  expanded  from  p  =  400,  v  =  2,  t  =  1200,  to  P  =  60,  V-  220. 
Find  the  final  temperature. 

13.  Along  the  path  PF1  2  =  c,  a  gas  is  expanded  to  ten  times  its  initial  volume  of 
10  cubic  feet  per  pound.     The  initial  pressure  being  1000,  and  the  value  of  It  53.36, 
find  the  final  pressure  and  temperature. 

14.  Through  what  range  of  temperature  will  air  be  heated  if  compressed  to  10  at- 
mospheres from  normal  atmospheric  pressure  and  70°  F.,  following  the  lawjw*-*  =  e? 
What  will  be  the  rise  in  temperature  if  the  law  is  pv"  =  c?    lfitispv=c? 

15.  Find  the  heat  imparted  to  one  pound  of  this  air  in  compressing  it  as  described 
according  to  the  \&w  pv1-8  =  c,  and  the  change  of  internal  energy. 

16.  In  Problem  14,  after  compression  along  the  path  pv1-8  =  c,  the  air  is  cooled 
at  constant  volume  to  70°  F.,  and  then  expanded  along  the  isodiabatic  path  to  its  initial 
volume.     Find  the  pressure  and  temperature  at  the  end  of  this  expansion. 

17.  The  isodiabatics  ab,  cd  are  intersected  by  lines  of  constant  volume  ac,  bd. 

Prove  — a  =  Q  and  ^  =^. 
PC      Pd          Tc^Td 

18.  In  a  room  at  normal  atmospheric  pressure  and  constant  temperature,  a 
cylinder  contains  air  at  a  pressure  of  1200  Ib.  per  square  inch.     The  stopcock  on  the 
cylinder  is  suddenly  opened.    After  the  pressure  in  the  cylinder  has  fallen  to  that  of 
the   atmosphere,  the  cock  is  closed,  and  the  cylinder  left  undisturbed  for  24  hours. 
Compute  the  pressure  in  the  cylinder  at  the  end  of  this  time. 

19.  Find  graphically  the  value  of  n  for  the  polytropic  curve  ab,  Fig.  41. 

20.  Plot  by  Brauer's  method  a  curve  pv1-*  =  26,200.     Use  a  scale  of  1  inch  per 
4  units  of  volume  and  per  80  units  of  pressure.     Begin  the  curve  with  p  =  1000. 

*  Absolute  pressures  are  pressures  measured  above  a  perfect  vacuum.     The  absolute 
pressure  of  one  standard  atmosphere  is  14.697  Ib.  per  square  inch. 


62  APPLIED  THERMODYNAMICS 

21.  Supply  the  necessary  figures  in  the  following  blank  spaces,  for  n  =  1.8,  and 
apply  the  results  to  check  the  curve    obtained    in   Problem  20.     Begin  with  v  =  6.12, 
p  =  1000. 

—  =  2.0,  2.25,  2.60,  3.0,  4.0,  5.0,  6.0,  7.0,  8.0 

v 

log-|=wlog  —  = 

JT  V 

p_  = 
p 

p  = 

22.  The  velocity  of  sound  in  air  being  taken  at  1140  ft.  per  second  at  70°  F.  and 
normal  atmospheric  pressure,  compute  the  value  of  y  for  air. 

23.  Compute  the  latent  heat  of  expansion  (Art.  58)  of  air  at  atmospheric  pressure 
and  32°  F. 

24.  Find  the  amount  of  heat  converted   into  work  in  a  cycle  1234,  in  which 
P1  =  P4  =  100,  FI  =  5,  F4  =  1,  P3  =  30,  and  the  equations  of  the  paths  are  as  follows  : 
for  41,  PV°  =  c:  for  12,  PVy  -  c ;  for  32,  PV=  c  ;  for  43,  PF1-8  =  c.     The  working 
substance  is  one  pound  of  air.     Find  the  temperatures  at  the  points  1,  2,  3,  4. 

25.  Find  the  exponent  of  the  poly  tropic  path,  for  air,  along  which  the  specific  heat 
is  —  k.     Also  that  along  which  it  is  —  I.    Represent  these  paths,  and  the  amounts  of  heat 
absorbed,  graphically,  comparing  with  those  along  which  the  specific  heats  are  k  and  Z, 
and  show  how  the  diagram  illustrates  the  meaning  of  negative  specific  heat. 


CHAPTER  VI 


THE   CARXOT   CYCLE 

128.  Heat  Engines.     In  a  heat  engine,  work  is  obtained  from 
heat  energy  through  the  medium  of  a  gas  or  vapor.     Of  the  total 
heat  received  by  such  fluid,  a  portion  is  lost  by  conduction  from  the 
walls  of  containing  vessels,  a  portion  is  discharged  to  the  atmosphere 
after  the  required  work  has  been  done,  and  a  third  portion  disap- 
pears, having  been  converted  into  external  mechanical  work.     By 
the  first  law  of  thermodynamics,  this  third  portion  is  equivalent  to 
the  work  done  ;  it  is  the  only  heat  actually  used.     The  efficiency  of  a 
heat  engine  is  the  ratio  of  the  net  heat  utilized  to  the  total  quantity  of 
heat  supplied  to  the  engine,  or,  of  external  work  done  to  gross  heat 

absorbed :  to  —  =  — ^— ,  in  which  h  denotes  the  quantity  of  heat 
If        H 

rejected  by  the  engine,  if  radiation  effects  be  ignored. 

129.  Cyclic  Action.     In  every  heat  engine,  the  working  fluid  passes 
through  a  series  of  successive  states  of  pressure,  volume,  and  temperature ; 
and,  in  order  that  operation  may  be  continuous,  it  is  necessary  either  that 
the  fluid  work  in  a  closed  cycle  which  may  be  repeated  indefinitely,  or 
that  a  fresh  supply  of  fluid  be  admitted  to  the  engine  to  compensate  for 
such  quantity  as  is  periodically 

discharged.  It  is  convenient  to 
regard  the  latter  more  usual  ar- 
rangement as  equivalent  to  the 
former,  and  in  the  first  instance 
to  study  the  action  of  a  constant 
body  of  fluid,  conceived  to  work 
continuously  in  a  closed  cycle. 


130.  Forms  of  Cycle.  The  sev- 
eral paths  described  in  Art.  83,  and 
others  less  commonly  considered,  sug- 
gest various  possible  forms  of  cycle, 
some  of  which  are  illustrated  in  Fig. 


FIG.  42.    Art.  130,  Problem  2.  — Possible  Cycles. 


42.    Many  of  these  have  been  given  names  (1).    The  isodiabatic  cycle,  bounded  by 
two  isothermals  and  any  two  isodiabatics  (Art.  108),  may  also  be  mentioned. 

63 


64  APPLIED  THERMODYNAMICS 

131.  Development  of  the  Carnot  Cycle.     Carnot,  in  1824,  by  describing  and 
analyzing  the  action  of  the  perfect  elementary  heat  engine,  effected  one  of  the 
most  important  achievements  of  modern  physical  science  (2).  Carnot,  it  is  true, 
worked  with  insufficient  data.    Being  ignorant  of  the  first  law  of  thermodynamics, 
and  holding  to  the  caloric  theory,  he  asserted  that  no  heat  was  lost  during  the 
cyclic  process ;  but,  though  to  this  extent  founded  on  error,  his  main  conclusions 
were  correct.     Before  his  death,  in  1832,  Carnot  was  led  to  a  more  just  conception 
of  the  true  nature  of  heat ;  while,  left  as  it  was,  his  work  has  been  the  starting 
point  for  nearly  all  subsequent  investigations.     The  Carnot  engine  is  the  limit 
and  standard  for  all  heat  engines. 

Clapeyron  placed  the  arguments  of  Carnot  in  analytical  and  graphical  form ; 
Clausius  expressed  them  in  terms  of  the  mechanical  theory  of  heat ;  James  Thomp- 
son, Rankine,  and  Clerk  Maxwell  corrected  Carnot's  assumptions,  redescribed  the 
cyclic  process,  and  redetermined  the  results ;  and  Kelvin  (3)  expressed  them  in 
their  final  and  satisfactory  modern  form. 

132.  Operation   of  Carnot's  Cycle.     Adopting  Kelvin's   method, 
the  operation  on  the  Carnot  engine  may  be  described  by  reference 
to  Fig.  43.     A  working  piston  moves  in  the  cylinder  c,  the  walls  of 

which  are  non-conduct- 
ing, while  the  head  is 
a  perfect  conductor. 
The  piston  itself  is 

FIG.  43.    Arts.  132,  138.  —  Operation  of  the  Carnot  Cycle.  ,  , 

a  non-conductor  and 

moves  without  friction.  The  body  8  is  an  infinite  source  of  heat 
(the  furnace,  in  an  actual  power  plant)  maintained  constantly  at 
the  temperature  T,  no  matter  how  much  heat  is  abstracted  from  it. 
At  r  is  an  infinite  condenser,  capable  of  receiving  any  quantity  of 
heat  whatever  without  undergoing  any  elevation  of  temperature 
above  its  initial  temperature  t.  The  plate /is  assumed  to  be  a  per- 
fect non-conductor.  The  fluid  in  the  cylinder  is  assumed  to  be 
initially  at  the  temperature  T  of  the  source. 

The  cylinder  is  placed  on  *.  Heat  is  received,  but  the  tempera- 
ture does  not  change,  since  both  cylinder  and  source  are  at  the 
same  temperature.  External  work  is  done,  as  a  result  of  the  recep- 
tion of  heat ;  the  piston  rises.  When  this  operation  has  continued 
for  some  time,  the  cylinder  is  instantaneously  transferred  to  the  non- 
conducting plate  /.  The  piston  is  now  allowed  to  rise  from  the  expan- 
sion produced  by  a  decrease  of  the  internal  energy  of  the  fluid.  It 
continues  to  rise  until  the  temperature  of  the  fluid  has  fallen  to  £, 


THE  CARNOT  CYCLE  65 

that  of  the  condenser,  when  the  cylinder  is  instantaneously  trans- 
ferred to  r.  Heat  is  now  given  up  by  the  fluid  to  the  condenser,  and 
the  piston  falls  ;  but  no  change  of  temperature  takes  place.  When  this 
action  is  completed  (the  point  for  completion  will  be  determined 
later),  the  cylinder  is  again  placed  on  /,  and  the  piston  allowed  to 
fall  further,  increasing  the  internal  energy  and  temperature  of  the 
gas  by  compressing  it.  This  compression  is  continued  until  the 
temperature  of  the  fluid  is  T  and  the  piston  is  again  in  its  initial 
position,  Avhen  the  cylinder  is  once  more  placed  upon  s  and  the  opera- 
tion may  be  repeated.  No  actual  engine  could  be  built  or  operated 
under  these  assumed  conditions. 

133.  Graphical    Representation.      The 
first  operation  described  in  the  preceding 
is  expansion  at  constant  temperature.     The 
path  of   the   fluid  is  then  an  isothermal. 
The  second  operation  is  expansion  without 
transfer  of  heat,  external  work  being  done 
at  the  expense    of   the  internal    energy ; 

the  path  is  consequently  adiabatic.  Dur-  FIG.  44.  Arts.  133-136, 138,  142. 
ing  the  third  operation,  we  have  isothermal  The  Carnot  Cycle' 

compression;  and  during  the  fourth,  adiabatic  compression.  The 
Carnot  cycle  may  then  be  represented  by  abed,  Fig.  44. 

134.  Termination  of  Third  Operation.    In  order  that  the  adiabatic  compression 
da  may  bring  the  fluid  back  to  its  initial  conditions  of  pressure,  volume,  and  tem- 
perature, the  isothermal  compression  cd  must  be  terminated  at  a  suitable  point  d. 
From  Art.  99, 

T     I  V  \i-'' 

—  =  ( —2  1       for  the  adiabatic  da, 
t       \Vd/ 

T      /  T'  \  i~v 

and  _  =  f  __» J        for  the  adiabatic  be ; 

t       \  \>  CJ 

hence  ii  =  -T-ft  and  ¥•  =  ¥*: 

Vd      Vc  V,      TV 

that  is,  the  ratio  of  volumes  during  isothermal  expansion  in  the  first  stage  must  be 
equal  to  the  ratio  of  volumes  during  isothermal  compression  in  the  third  stage,  if  the 
final  adiabatic  compression  is  to  complete  the  cycle.  (Compare  Art.  108.) 

135.  Efficiency  of  Carnot  Cycle.     The  only  transfers  of  heat  dur- 
ing this  cycle  occur  along  ab  and  cd.     The  heat  absorbed  along  ab  is 


66  APPLIED  THERMODYNAMICS 

77 


e-^-     Similarly,  along   cd,  the  heat  rejected 

a  'a 

is  Rt  loge  —  -     The  net  amount  of  heat  transformed  into  work  is  the 

difference  of  these  two  quantities  ;  whence  the  efficiency,  defined  in 
Art.  128  as  the  ratio  of  the  net  amount  of  heat  utilized  to  the  total 
amount  of  heat  absorbed,  is 


',  since  -~-  =  -~r^  from  Art.  134. 

*  a          *  d 


136.    Second  Derivation.     The  external  work  done  under  the  two  adiabatics 
be,  da  is 

pbvb-pcvc  and  pava  -  pdvd 
y-i  y-1 

Deducting  the  negative  work  from  the  positive,  the  net  adiabatic  work  is 


but  PaVa  =  PbVb,  from  the  law  of  the  isothermal  a/>;  similarly,  PdVd  =  PCVC>  and 
consequently  this  network  is  equal  to  zero;  and  if  we  express  efficiency  by  the 
ratio  of  work  done  to  gross  heat  absorbed,  we  need  consider  only  the  work  areas 
under  the  isothermal  curves  cib  and  cd,  which  are  given  by  the  numerator  in  the 
expression  of  Art.  135. 

The  efficiency  of  the  Carnot  engine  is  therefore  expressed  by  the 
ratio  of  the  difference  of  the  temperatures  of  source  and  condenser  to 
the  absolute  temperature  of  the  source. 

137.  Carnot's   Conclusion.     The   computations  described   apply  to   any  sub- 
stance  in    uniform   thermal   condition  ;    hence   the   conclusion,  now  universally 
accepted,  that  the  motive  power  of  heat  is  independent  of  the  agents  employed  to 
develop  it  ;  it  is  determined  solely  by  the  temperatures  of  the  bodies  between  which 
the  cyclic  transfers  of  heat  occur. 

138.  Reversal  of  Cycle.     The  paths  which  constitute  the  Carnot  cycle, 
Fig.  44,  are  poly  tropic  and  reversible  (Art.  125);  the  cycle  itself  is  rever- 
sible.    Let  the  cylinder  in  Fig.  43  be  first  placed  upon  r,  and  the  piston 
allowed  to  rise.     Isothermal  expansion  occurs.     The  cylinder  is  trans- 
ferred to  /and  the  piston  caused  to  fall,  producing  adiabatic  compression. 
The  cylinder  is  then  placed  on  s,  the  piston  still  falling,  resulting  in  iso- 
thermal compression;  and  finally  on/,  the  piston  being  allowed  to  rise,  so 
as  to  produce  adiabatic  expansion.     Heat  has  now  been  taken  from  the 


REVERSIBILITY  67 

condenser  and  rejected  to  the  source.  The  cycle  followed  is  dcbad,  Fig.  44. 
Work  has  been  expended  upon  the  fluid ;  the  heat  delivered  to  the  source  s  is 
made  up  of  the  heat  taken  from  the  condenser  r,  plus  the  heat  equivalent  of 
the  work  done  upon  the  fluid.  The  apparatus,  instead  of  being  a  heat 
engine,  is  now  a  sort  of  heat  pump,  transferring  heat  from  a  cold  body  to 
one  warmer  than  itself,  by  reason  of  the  expenditure  of  external  work. 
Every  operation  of  the  cycle  has  been  reversed.  The  same  quantity  of 
heat  originally  taken  from  s  has  now  been  given  up  to  it ;  the  quantity 
of  heat  originally  imparted  to  r  is  now  taken  from  it ;  and  the  amount  of 
external  work  originally  done  by  the  fluid  has  now  been  expended  upon 
it.  The  efficiency,  based  on  our  present  definition,  may  exceed  unity ;  it 
is  the  quotient  of  heat  imparted  to  the  source  by  work  expended.  The 
cylinder  c  must  in  this  case  be  initially  at  the  temperature  t  of  the  con- 
denser r. 

139.  Criterion  of  Reversibility.     Of   all   engines  working  between   the 
same  limits  of  temperature,  that  which  is  reversible  is  the  engine  of  maximum 
efficiency. 

If  not,  let  A  be  a  more  efficient  engine,  and  let  the  power  which  this 
engine  develops  be  applied  to  the  driving  of  a  heat  pump  (Art.  138), 
(which  is  a  reversible  engine),  and  let  this  heat  pump  be  used  for  restor- 
ing heat  to  a  source  s  for  operating  engine  A.  Assuming  that  there  is  no 
friction,  then  engine  A  is  to  perform  just  a  sufficient  amount  of  work  to 
drive  the  heat  pump.  In  generating  this  power,  engine  A  will  consume 
a  certain  amount  of  heat  from  the  source,  depending  upon  its  efficiency. 
If  this  efficiency  is  greater  than  that  of  the  heat  pump,  the  latter  will  dis- 
charge more  heat  than  the  former  receives  (see  explanation  of  efficiency, 
Art.  138) ;  or  will  continually  restore  more  heat  to  the  source  than  engine 
A  removes  from  it.  This  is  a  result  contrary  to  all  experience.  It  is 
impossible  to  conceive  of  any  self-acting  machine  which  shall  continually 
produce  heat  (or  any  other  form  of  energy)  without  a  corresponding  con- 
sumption of  energy  from  some  other  source. 

140.  Hydraulic  Analogy.     The  absurdity  may  be  illustrated,  as  by  Heck  (4), 
by  imagining  a  water  motor  to  be  used  in  driving  a  pump,  the  pump  being  em- 
ployed to  deliver  the  water  back  to  the  upper  level  which  supplies  the  motor. 
Obviously,  the  motor  would  be  doing  its  best  if  it  consumed  no  more  water  than 
the  pump  returned  to  the  reservoir ;  no  better  performance  can  be  imagined,  and 
with  actual  motors  and  pumps  this  performance  would  never  even  be  equaled. 
Assuming  the  pump  to  be  equally  efficient  as  a  motor  or  as  a  pump  (i.e.  reversible), 
the  motor  cannot  possibly  be  more  efficient. 

141.  Clausius*  Proof.     The  validity  of  this  demonstration  depends  upon  the 
correctness  of  the  assumption  that  perpetual  motion  is  impossible.     Since  the  im- 


68  APPLIED  THERMODYNAMICS 

possibility  of  perpetual  motion  cannot  be  directly  demonstrated,  Clausius  estab- 
lished the  criterion  of  reversibility  by  showing  that  the  existence  of  a  more  effi- 
cient engine  A  involved  the  continuous  transference  of  heat  from  a  cold  body  to 
one  warmer  than  itself,  without  the  aid  of  external  agency :  an  action  which  is  axio- 
matically  impossible. 

142.  The  Perfect  Elementary  Heat  Engine.     It  follows  from  the  analysis  of 
Art.  135  that  all  engines  working  in  the  Carnot  cycle  are  equally  efficient ;  and 
from  Art.  139  that  the  Carnot  engine  is  one  of  that  class  of  engines  of  highest  effi- 
ciency.    The  Carnot  cycle  is  therefore  described  as  that  of  the  perfect  elementary 
heat  engine.     It  remains  to  be  shown  that  among  reversible  engines  working  be- 
tween equal  temperature  limits,  that  of  Carnot  is  of  maximum  efficiency.     Con- 
sider the  Carnot  cycle  abed,  Fig.  44.     The  external  work  done  is  abed,  and  the 
efficiency,  abed  -4-  nabN.     For  any  other  reversible  path  than  ab,  like  ae  or  fb, 
touching  the  same  line  of  maximum  temperature,  the  work  area  abed  and  the  heat 
absorption  area  nabN  are  reduced  by  equal  amounts.     The  ratio  expressing  effi- 
ciency is  then  reduced  by  equal  amounts  in  numerator  and  denominator,  and  since 
the  value  of  this  ratio  is  always  fractional,  its  value  is  thus  always  reduced.     For 
any  other  reversible  path  than  cd,  like  ch  or  gd,  touching  the  same  line  of  mini- 
mum temperature,  the  work  area  is  reduced  without  any  reduction  in  the  gross 
heat  area  nabN.     Consequently  the  Carnot  engine  is  that  of  maximum  efficiency 
among  all  conceivable  engines  worked  between  the  same  limits  of  temperature.     A 
practical  cycle  of  equal  efficiency  will,  however,  be  considered  (Art.  257). 

143.  Deductions.     The  efficiency  of  an  actual    engine   can   therefore 
never  reach  100  per  cent,  since  this,  even  with  the  Carnot  engine,  would 
require  t  in  Art.  135  to  be  equal  to  absolute  zero.     High  efficiency  is  con- 
ditioned upon  a  wide  range  of  working  temperatures ;  and  since  the  mini- 
mum temperature  cannot  be  maintained  below  that  of  surrounding  bodies, 
high  efficiency  involves  practically  the  highest  possible  temperature  of 
heat  absorption.     Actual  heat  engines  do  not  work  in  the  Carnot  cycle ; 
but  their  efficiency  nevertheless  depends,  though  less  directly,  on  the  tem- 
perature range.     With  many  working  substances,  high  temperatures  are 
necessarily  associated  with  high  specific  pressures,  imposing  serious  con- 
structive difficulties.     The  limit  of  engine  efficiency  is  thus  fixed  by  the 
possibilities  of  mechanical  construction. 

(1)  Alexander,  Treatise  on  Thermodynamics,  1893,  38-40.  (2)  Carnot's  Reflec- 
tions is  available  in  Thurston's  translation  or  in  Magie's  Second  Law  of  Thermody- 
namics. An  estimate  of  his  part  in  the  development  of  physical  science  is  given  by 
Tait,  Thermodynamics,  1868,  44.  (3)  Trans.  Roy.  Soc.  Edinburgh,  March,  1851  ; 
Phil.  Mag.,  IV,  1852  ;  Math,  and  Phys.  Papers,  I,  174.  (4)  The  Steam  Engine,  I, 
50. 

SYNOPSIS   OF   CHAPTER   VI 

Heat  engines  :  efficiency  =  heat  utilized  -f-  heat  absorbed  —  =  — • 

Cyclic  action  :  closed  cycle  ;  forms  of  cycle. 


THE  CARNOT   CYCLE  69 

Carnot  cycle  :  historical  development  ;  cylinder,  source,  insulating  plate,  condenser  ; 
graphical  representation  ;   termination  of  third  operation,  when  —  ^  =  -Z?  ;    effi- 


Carnot'  s  conclusion  :  efficiency  is  independent  of  the  working  substance.  v 

Reversal  of  cycle  :  the  reversible  engine  is  that  of  maximum   efficiency  ;  hydraulic 

analogy. 

Carnot  cycle  not  surpassed  in  efficiency  by  any  reversible  or  irreversible  cycle. 
Limitations  of  efficiency  in  actual  heat  engines. 


PROBLEMS 

1.  Show  how  to  express  the  efficiency  of  any  heat-engine  cycle  as  the  quotient 
of  two  areas  on  the  PV diagram. 

2.  Draw  and  explain  six  forms  of  cycle  not  shown  in  Fig.  42. 

3.  In  a  Carnot  cycle,  using  air,  the  initial  state  is  P  =  1000,  V  =  100.     The  pres- 
sure after  isothermal  expansion  is  500,  the  temperature  of  the  condenser  200°  F.     Find 
the  pressure  at  the  termination  of  the  "third  operation,"  the  external  work  done  along 
each  of  the  four  paths,  and  the  heat  absorbed  along  each  of  the  four  paths.     Units  are 
cubic  feet  per  pound  and  pounds  per  square  foot. 

4.  A  non-reversible  heat  engine  takes  1  B.  t.  u.  per  minute  from  a  source  and  is 
used  to  drive  a  heat  pump  having  an  efficiency  (quotient  of  work  by  heat  imparted  to 
source)  of  0.70.     What  would  be  the  rate  of  increase  of  heat  contents  of  the  source  if 
the  efficiency  of  the  heat  engine  were  0.80  ? 

5.  Ordinary  non-condensing  steam  engines  use  steam  at  325°  F.  and  discharge  it 
to  the  atmosphere  at  215°  F.     What  is  their  maximum  possible  efficiency  ? 

6.  Find  the  limiting  efficiency  of  a  gas  engine  in  which  a  maximum  temperature 
of  3000°  F.  is  attained,  the  gases  being  exhausted  at  1000°  F. 

7.  An  engine  consumes  225  B.  t.  u.  per  indicated  horse  power  (33,000  foot-pounds) 
per  minute.     If  its  temperature  limits  are  430°  F.  and  105°  F.,  how  closely  does  its 
efficiency  approach  the  best  possible  efficiency  ? 


CHAPTER   VII 

THE   SECOND   LAW   OF   THERMODYNAMICS 

144.  Statement  of  Second  Law.  The  expression  for  efficiency  of 
the  Carnot  cycle,  given  in  Art.  135,  is  a  statement  of  the  second  law 
of  thermodynamics.  The  law  is  variously  expressed  ;  but,  in  general, 
it  is  an  axiom  from  which  is  established  the  criterion  of  reversibility 
(Art.  139). 

With  Clausius,  the  axiom  was, 

(a)  "  Heat  cannot  of  itself  pass  from  a  colder  to  a  hotter  body;  "  while  the 
equivalent  axiom  of  Kelvin  was, 

(6)  "  It  is  impossible,  by  means  of  inanimate  material  agency,  to  derive 
mechanical  effect  from  any  portion  of  matter  by  cooling  it  below  the  tempera- 
ture of  the  coldest  of  surrounding  objects" 

With  Carnot,  the  axiom  was  that  perpetual  motion  is  impossible;  while  Ran- 
kine's  statement  of  the  second  law  (Art.  151)  is  an  analytical  restatement  of  the 
efficiency  of  the  Carnot  cycle. 

145.  Comparison  of  Laws.  The  law  of  relation  of  gaseous  properties  (Art.  10) 
and  the  second  law  of  thermodynamics  are  justified  by  their  results,  while  the  first 
law  of  thermodynamics  is  an  expression  of  experimental  fact.  The  second  law  is  a 
"  definite  and  independent  statement  of  an  axiom  resulting  from  the  choice  of  one 
of  the  two  propositions  of  a  dilemma"  (1).  For  example,  in  Carnot's  form,  we 
must  admit  either  the  possibility  of  perpetual  motion  or  the  criterion  of  reversi- 
bility ;  and  we  choose  to  admit  the  latter.  The  second  law  is  not  a  proposition  to 
be  proved,  but  an  "  axiom  commanding  universal  assent  when  its  terms  are 
understood." 

146.  Preferred  Statements.  The  simplest  and  most  satisfactory  statement  of 
the  second  law  may  be  derived  directly  from  inspection  of  the  formula  for  effi- 
ciency, (T  —  t)  H-  T  (Art.  135).  The  most  general  statement, 

(c)  "  The  availability  of  heat  for  doing  work  depends  upon  its  temperature"  leads 
at  once  to  the  axiomatic  forms  of  Kelvin  and  Clausius;  while  the  most  specific  of 
all  the  statements  directly  underlies  the  presentation  of  liankine : 

(d)  "  If  all  of  the  heat  be  absorbed  at  one  temperature,  and 
rejected  at  another  lower  temperature,  the  heat  transformed  to 

70 


THE   SECOND   LAW   OF  THERMODYNAMICS  71 

external  work  is  to  the  total  heat  absorbed  in  the  same  ratio  as  that 
of  the  difference  between  the  temperatures  of  absorption  and  rejec- 
tion to  the  absolute  temperature  of  absorption  ;"  or, 


%      H  T 

in  which  H  represents  heat  absorbed  ;  and  A,  heat  rejected. 

147.  Other  Statements.     Forms  (a),  (ft),  (c),  and  (d)  are  those  usually  given 
the  second  law.     In  modified  forms,  it  has  been  variously  expressed  as  follows : 

(e)  "  All  reversible  engines  working  between  the  same  uniform  tem- 
peratures have  the  same  efficiency." 

(/)  "  The  efficiency  of  a  reversible  engine  is  independent  of  the  nature 
of  the  working  substance." 

(g)  "  It  is  impossible,  by  the  unaided  action  of  natural  processes, 
to  transform  any  part  of  the  heat  of  a  body  into  mechanical  work,  except 
by  allowing  the  heat  to  pass  from  that  body  into  another  at  lower 
temperature." 

(h)  "If  the  engine  be  such  that,  when  it  is  worked  backward,  the 
physical  and  mechanical  agencies  in  every  part  of  its  motions  are  reversed, 
it  produces  as  much  mechanical  effect  as  can  be  produced  by  any  therm  o 
dynamic  engine,  with  the  same  source  and  condenser,  from  a  given  quan- 
tity of  heat." 

148.  Harmonization  of  Statements.     It  has  been  asserted  that  the  state- 
ments of  the  second  law  by  different  writers  involve  ideas  so  diverse  as, 
apparently,  not  to  cover  a  common  principle.     A  moment's  consideration 
of  Art.  144  will  explain  this.     The  second  law,  in  the  forms  given  in  (a), 
(6),  (c),  (g),  is  an  axiom,  from  which  the  criterion  of  reversibility  is  estab- 
lished.    In  (d),  (e)  (/),  it  is  a  simple  statement  of  the  efficiency  of  the  Car- 
not   cycle,  with  which   the   axiom   is  associated ;  while  in  (7i),  it  is  the 
criterion  of  reversibility  itself.     Confusion   may  be  avoided  by  treating 
the  algebraic  expression  of  (d),  Art.   146,   as   a   sufficient  statement  of 
the  second  law,  from  which  all  necessary  applications  may  be  derived. 

149.  Consequences  of  the  Second  Law.     Some  of  these  were  touched  upon  in 
Art.  143.     The  first  law  teaches  that  heat  and  work  are  mutually  convertible, 
the  second  law  shows  how  much  of  either  may  be  converted  into  the  other  under 
stated  conditions.     Ordinary  condensing  steam  engines  work  between   tempera- 
tures  of   about  350°  F.  and  100°  F.     The  maximum  possible  efficiency  of  such 
engines  is  therefore 

350  -  100 
350  +  459.4 


72  APPLIED  THERMODYNAMICS 

The  efficiencies  of  actual  steam  engines  range  from  1\  to  25  per  cent,  with  an 
average  probably  not  exceeding  7  to  10  per  cent.  A  steam  engine  seems  therefore 
a  most  inefficient  machine  ;  but  it  must  be  remembered  that,  of  the  total  heat 
supplied  to  it,  a  large  proportion  is  (by  the  second  law)  unavailable  for  use,  and 
must  be  rejected  when  its  temperature  falls  to  that  of  surrounding  bodies.  We  can- 
not expect  a  water  wheel  located  in  the  mountaius^to  utilize  all  of  the  head  of  the 
water  supply,  measured  down  to  sea  level.  The  available  head  is  limited  by  the 
elevation  of  the  lowest  of  surrounding  levels.  The  performance  of  a  heat  engine 
should  be  judged  by  its  approach  to  the  efficiency  of  the  Carnot  cycle,  rather  than 
by  its  absolute  efficiency. 

Heat  must  be  regarded  as  a  "low  unorganized"  form  of  energy,  which  pro- 
duces useful  work  only  by  undergoing  a  fall  of  temperature.  All  other  forms  of 
energy  tend  to  transform  themselves  into  heat.  As  the  universe  slowly  settles  to 
thermal  equilibrium,  the  performance  of  work  by  heat  becomes  impossible  and  all 
energy  becomes  permanently  degenerated  to  its  most  unavailable  form. 


150.   Temperature  Fall  and  Work  Done.     Consider  the  Carnot  cycle, 

Fig.  45,  the  total  heat  absorbed  being  nabN  and  the  efficiency  abed  -f-  nabN 

=  (T—t)-%-T.  Draw  the  isothermals 
efj  gh,  ij,  successively  differing  by  equal 
temperature  intervals  ;  and  let  the  tem- 
peratures of  these  isothermals  be  Tlt 
T2,  TV  Then  the  work  done  in  cycle 
abfe  is  nabNx(T—  TJ-i-  T7;  that  in 
cycle  abhg  is  nabNx  (T—  T2)-*-T;  that 
in  cycle  abji  is  nabN  x  (  T  —  Ts)  -f-  T. 
As  (T-TS)  =  3(T-  T,)  and  (T-T2) 
=  2  (  T  -  7\)  ,  abji  =  3(abfe)  and  abhg 
=  2  (abfe);  whence  abfe  =  efhg  =  gliji. 

FIG.  45.  Arts.  150,  153,  154,  m-Second  In  other  words,  the  external  work 

available  from  a  definite  temperature  fall 

is  the  same  at  all  parts  of  the  thermometric  scale.    The  waterfall  analogy  of 

Art.  149  may  again  be  instructively  utilized. 

151.  Rankine's  Statement  of  the  Second  Law.    "If  the  total  actual  heat  of  a 
uniformly  hot  substance  be  conceived  to  be  divided  into  any  number  of  equal  parts,  the 
effects  of  those  parts  in  causing  work  to  be  performed  are  equal."     If  \ve  remember 
that  by  "  total  actual  heat  "  Rankine  means  the  heat  corresponding  to  absolute  tem- 
perature, his  terse  statement  becomes  a  form  of  that  just  derived,  dependent  solely 
upon  the  computed  efficiency  of  the  Carnot  cycle. 

152.  Absolute  Temperature.     It  is  convenient  to  review  the  steps  by  which 
the  proposition  of  Art.  150  has  been  established.     We  have  derived  a  conception 
of  absolute  temperature  from  the  law  of  Charles,  and  have  found  that  the  effi- 
ciency of  the  Carnot  cycle  bears  a  certain  relation  to  definite  absolute  temperatures. 


KELVIN'S  ABSOLUTE  SCALE  73 

Our  scale  of  absolute  temperatures,  practically  applied,  is  not  entirely  satisfactory  ; 
for  the  absolute  zero  of  the  air  thermometer,  —  459.4°  F.,  is  not  a  true  absolute 
zero,  because  air  is  not  a  perfect  gas.  The  logical  scale  of  absolute  temperature 
would  be  that  in  which  temperatures  were  defined  by  reference  to  the  work  done 
by  a  reversible  heat  engine.  Having  this  scale,  we  should  be  in  a  position  to  com- 
pute the  coefficient  of  expansion  of  a  perfect  gas. 

153.  Kelvin's  Scale  of  Absolute  Temperature.  Kelvin,  in  1848,  was  led 
by  a  perusal  of  Carnot's  memoir  to  propose  such  a  scale.  His  first  defini- 
tion, based  on  the  caloric  theory,  resulted  only  in  directing  general  atten- 
tion to  Carnot's  great  work  ;  his  second  definition  is  now  generally  adopted. 
Its  form  is  complex,  but  the  conception  involved  is  simply  that  of  Art.  150: 

"  The  absolute  temperatures  of  two  bodies  are  proportional  to  the  quanti- 
ties of  heat  respectively  taken  in  and  given  out  in  localities  at  one  temperature 
and  at  the  other,  respectively,  by  a  material  system  subjected  to  a  complete 
cycle  of  perfectly  reversible  thermodynamic  operations,  and  not  allowed  to  part 
with  or  take  in  heat  at  any  other  temperature."  Briefly, 

"  The  absolute  values  of  two  temperatures  are  to  each  other  in  the  propor- 
tion of  the  quantities  of  heat  taken  in  and  rejected  in  a  perfect  thermodynamic 
engine,  working  with  a  source  and  condenser  at  the  higher  and  the  lower  of 
the  temperatures  respectively."  Symbolically, 

T     H  .    T 

-  =      5  or>  m  Fig-  45,  — 


, 
t       h  t      ndcN 

This  relation  may  be  obtained  directly  by  a  simple  algebraic  trans- 
formation of  the  equation  for  the  second  law,  given  in  Art.  146,  (d). 

154.  Graphical  Representation  of  Kelvin's  Scale.     Keturning  to  Fig.  45, 
but  ignoring  the  previous  significance  of  the  construction,  let  ab  be  an  iso- 
thermal and  an,  bl?  adiabatics.     Draw  isothermals  ef,  gh,  ij,  such  that  the 
areas  abfe,  efhg,  ghji  are  equal.     Then  if  we  designate  the  temperatures 
along  ab,  ef,  gh,  ij  by   T,  T19  T,,  Ts,  the  temperature  intervals  T-  Tl9 
T7!-  T2,  T2-  T3  are  equal.     If  we  take  ab  as  212°  F.,  and  cd  as  32°  F., 
then  by  dividing  the  intervening  area  into  180  equal  parts,  we  shall  have 
a  true  Fahrenheit  absolute  scale.     Continuing  the  equal  divisions  down 
below  cd,  we  should  reach  a  point  at  which  the  last  remaining  area  be- 
tween the  indefinitely  extended  adiabatics  was  just  equal  to  the  one  next 
preceding,  provided  that  the  temperature  32°F.  could  be  expressed  in  an 
even  number  of  absolute  degrees. 

155.  Carnot's  Function.     Carnot  did  not  find   the  definite  formula  for  effi- 
ciency of  his  engine,  given  in  Art.  135,  although  he  expressed  it  as  a  function  of 
the  temperature  range  (T-t).     We  may  state  the  efficiency  as 


74  APPLIED  THERMODYNAMICS 

z  being  a  factor  having  the  same  value  for  all  gases.     Taking  the  general  expres- 

TT  J 

sion  for  efficiency,  -       *  (Art.  128),  and  making  H  =  h  +  dh}  we  have 
H 

h  +  dh  —  h         dh 


h  +  dh         h  +  dh 

For  e  =  z(T  —  t),  we  may  write  e  =  zdt  or  z  —  •— ,  giving 

at 

z  =     dh    -  4-  dt,  equivalent  to  — . 
h  -f  dh  hdt 

But  -  =  —  (Art.  153)  ;  whence  ^-±-^  =  h  +  dh  and  —  =  — ,  and  *  =  —  =  1. 
t       h  t  h  t        h  dh      z 

-I  rp  i          rp  j 

Then  z  =  -  and  e  = =  — - —  in  finite  terms;  as  already  found.     The  factor  z 

is  known  as  Carnot's  function.     It  is  the  reciprocal  of  the  absolute  temperature. 

156.  Determination  of  the  Absolute  Zero.  The  porous  plug  experiments  "con- 
ducted by  Joule  and  Kelvin  (Art.  74)  consisted  in  forcing  various  gases  slowly 
through  an  orifice.  The  fact  has  already  been  mentioned  that  when  this  action 
was  conducted  without  the  performance  of  external  work,  a  barely  noticeable 
change  in  temperature  was  observed ;  this  being  with  some  gases  an  increase,  and 
with  others  a  decrease.  When  a  resisting  pressure  was  applied  at  the  outlet  of  the 
orifice,  so  as  to  cause  the  performance  of  some  external  work  during  the  flow  of 
gas,  a  fall  of  temperature  was  observed  ;  and  this  fall  ivas  different  for  different  gases. 

The  "  porous  plug  "  was  a  wad  of  silk  fibers  placed  in  the  orifice  for  the  purpose 
of  reconverting  all  energy  of  velocity  back  to  heat.  Assume  a  slight  fall  of  tem- 
perature to  occur  in  passing  the  plug,  the  velocity  energy  being  reconverted  to 
heat  at  the  decreased  temperature,  giving  the  equivalent  paths  ad,  dc,  Fig.  45. 
Then  expend  a  sufficient  measured  quantity  of  work  to  bring  the  substance  back 
to  its  original  condition  a,  along  cba.  By  the  second  law, 

T     _     T}    _          T,  and  T_  _        ncbN 

nabN  ~  nefN  ~  nabN  -  abfe '  *        J\~  nabN  -  abfe' 

or  T-T  =  T    (       n°bN       _{\_T          (*hfe 

1   \nabN  -  abfe        )        !  nabN  -  abfe 

If  (T—  7\)  as  determined  by  the  experiment  =  a,  and  nabN  be  put  equal  to  unity, 

T  =a(l-al>fe)j 

in  which  abfe  is  the  work  expended  in  bringing  the  gas  back  to  its  original  tem- 
perature. This,  in  outline,  was  the  Joule  and  Kelvin  method  for  establishing  a 
location  for  the  true  absolute  zero:  the  complete  theory  is  too  extensive  for  pres- 
entation here  (2).  The  absolute  temperature  of  melting  ice  is  on  this  scale 
491. 58°  F.  or  273.1°  C. 

The  agreement  with  the  hydrogen  or  the  air  thermometer  is  close. 
The  correction  for  the  former  is  generally  less  than  T^°  C.,  and  that  for 


THE   SECOND   LAW  OF  THERMODYNAMICS  75 

the  latter  less  than  Ty  C.  Wood  has  computed  (3)  that  the  true  absolute 
zero  must  necessarily  be  slightly  lower  than  that  of  the  air  thermometer. 
According  to  Alexander,  (4)  the  difference  of  the  two  scales  is  constant  for 
all  temperatures.  The  Kelvin  absolute  scale  establishes  a  logical  defini- 
tion of  temperature  as  a  physical  unit.  Actual  gas  thermometer  tempera- 
tures may  be  reduced  to  the  Kelvin  scale  as  a  final  standard. 

In  the  further  discussion,  the  temperature  —459.6°  F.  will  be  regarded 
as  the  absolute  zero. 

(1)  Peabody,     Thermodynamics,    1907,    27.       (2)  Phil.    Trans.,    CXLIV,    349. 
(3)    Thermodynamics,  1905,  116.     (4)   Treatise  on  Thermodynamics,  1892,  91. 


SYNOPSIS   OF   CHAPTER   VII 

Statements  of  the  second  law  :    an  axiom  establishing  the  criterion  of  reversibility  ; 
H—  h  _  T  —  t  QT  h  _JL       a  statement  of  the  efficiency  of  the  Carnot  cycle  ;  the  cri- 

H  T          H  ~  T      terion  of  reversibility  itself. 

The  second  law  limits  the  possible  efficiency  of  a  heat  engine. 
The  fall  of  temperature  determines  the  amount  of  external  work  done. 
Temperature  ratios  defined  as  equal  to  ratios  of  heats  absorbed  and  emitted. 
The  Carnot  function  for  cyclic  efficiency  is  the  reciprocal  of  the  absolute  temperature. 
The  absolute  zero,  based  on  the  seconl  law,  is  at  —  459.6°  F. 


PROBLEMS 

1.  Illustrate  graphically  the  first  and  the  second  laws  of  thermodynamics.    Frame 
a  new  statement  of  the  latter. 

2.  An  engine  works  in  a  Carnot  cycle  between  400°  F.  and  280°  F.,  developing 
120  h.p.     If  the  heat  rejected  by  this  engine  is  received  at  the  temperature  of  rejection 
by  a  second  Carnot  engine,  which  works  down  to  220°  F.,  find  the  horse  power  of  the 
second  engine. 

3.  Find  the  coefficient  of  expansion  at  constant  pressure  of  a  perfect  gas.     What 
is  the  percentage  difference  between  this  coefficient  and  that  for  air  ? 

4.  A  Carnot  engine  receives  from  the  source  1000  B.  t.  u.,  and  discharges  to  the 
condenser  500  B.  t.  u.     If  the  temperature  of  the  source  is  600°  F.,  what  is  the  tem- 
perature of  the  condenser  ? 

5.  A  Carnot  engine  receives  from  the  source  190  B.  t.  u.  at  1440.4°  F.,  and  dis- 
charges to  the  condenser  90  B.  t.  u.  at  440.4°  F.     Find  the  location  of  the  absolute 


6.  In  the  porous  plug  experiment,  the  initial  temperature  of  the  gas  being  that  of 
melting  ice,  and  the  fall  of  temperature  being  ^  of  the  range  from  melting  to  boiling 
of  water  at  atmospheric  pressure,  the  work  expended  in  restoring  the  initial  tempera- 
ture was  1.58  foot  pounds.  Find  the  absolute  temperature  at  32°  F. 


76  APPLIED  THERMODYNAMICS 


REVIEW  PROBLEMS,  CHAPTERS   I-VII 

1.    State  the  precise  meaning,  or  the  application,  of  the  following  expressions  : 
k  I  y  R  I       778         P 

dH  V  PV-PV  PV-PV 


/P\        =T 

\pl  t 


pvn  =  c 


2.  A  heat  engine  receives  its  fluid  at  350°  F.  and  discharges  it  at  110°  F.    It  con- 
sumes 200  B.  t.  u.  per  Ihp.  per  minute.     Find  its  efficiency  as  compared  with  that  of 
the  corresponding  Carnot  cycle. 

3.  Given  a  cycle  o6c,  in  which  Pa  =  Pb  =  100  Ib.  per  sq.  in.,  Va  =  1,    ~  =  6, 

'a 


=  PcFe1-8,  PaVa  =  PCFC,  find  the  pressure,  volume,  and  temperature  at  c  if  the 
substance  is  1  Ib.  of  air. 

4.  Find  the  pressure  of  100  Ib.  of  air  contained  in  a  100  cu  -ft.  tank  at  82°  F. 

5.  A  heat  engine  receives  1175.2  B.  t.  u.  in  each  pound  of  steam  and  rejects 
1048.4  B.  t.  u.     It  uses  3110  Ib.  of  steam  per  hour  and  develops  142  hp.     Estimate  the 
value  of  the  mechanical  equivalent  of  heat. 

6.  One  pound  of  air  at  32°  F.  is  compressed  from  14.7  to  2000  Ib.  per  square  inch 
without  change  of  temperature.     Find  the  percentage  change  of  volume. 

rr>  _    j. 

7.  Prove  that  the  efficiency  of  the  Carnot  cycle  is 


8.  Air  is  heated  at  constant  pressure  from  32°  F.  to  500°  F.    Find  the  percentage 
change  in  its  specific  volume. 

9.  Prove  that  the  change  of  internal  energy  in  passing  from  a  to  b  is  independent 
of  the  path  ab. 

10.  Given  the  formula  for  change  of  internal  energy,      6    fc~  —  ^—  ?  ,  prove  that 
2&-J0B  =  1(7*6  -T.). 

11.  Given  It  for  air  =  53.36,  F  =  12.387  ;  and  given  V=  178.8,  k  —  3.4  for  hydro- 
gen :  find  the  value  of  y  for  hydrogen. 

12.  Explain  isothermal,  adiabatic,  isodynamic,  isodiabatic. 

13.  Find  the  mean  specific  heat  along  the  path  pv1-*  =  c  for  air  (Z  =  0.1689). 

14.  A  steam  engine  discharging  its  exhaust  at  212°  F.  receives  steam  containing 
1100  B.  t.  u.  per  pound  at  500°  F.     What  is  the  minimum  weight  of  steam  it  may  use 
per  Ihp.-hr.  ? 

15.  A  cycle  is  bounded  by  polytropic  paths  12,  23,  13.     We  have  given 

Pl  =  P2  =  100,000  Ibs.  per  sq.  ft. 
F2  =  F3  =  40  cubic  feet  per  pound. 
7\  =  3000°  F. 
P,.Fi  =  P3F8. 
Find  the  amount  of  heat  converted  to  work  in  the  cycle. 


CHAPTER  VIII 


ENTROPY 

157.  Adiabatic  Cycles.    Let  abdc,  Fig.  46,  be  a  Carnot  cycle,  an  and  bN 
the  projected  adiabatics.     Draw  intervening  adiabatics  em,  gM,  etc.,  so 
located  that  the  areas  naem,  megM,  MgbN,  are  equal.     Then  since  the  effi- 
ciency of  each  of  the  cycles  aefc,  eghf,  gbdh,  is  (T  —  t)  -5-  T,  the  work  areas 
represented  by  these  cycles  are  all  equal.    To  measure  these  areas  by  mechani- 
cal means  would  lead  to  approximate  results  only. 

158.  Rectangular   Diagram.     If   the  adiabatics  and  isothermals 
were  straight  lines,  simple  arithmetic  would  suffice  for  the  measure- 
ment of  the  work  areas  of  Fig.  46.     We 

have  seen  that  in  the  Carnot  cycle, 
bounded  by  isothermals  and  adiabatics, 

—  =  —  (Art.  153).  Applying  this  for- 
mula to  Rankine's  theorem  (Art.  106), 
we  have  the  quotient  of  an  area  and  a 
length  as  a  constant.  If  the  area  h  is 
a  part  of  ZT,  then  there  must  be  some 
constant  property,  which,  when  multi- 


plied  by  the  temperatures   T  or  £,    will 


FIG.  47.    Arts.  158,   163,   171.  — En- 
tropy Diagram. 


FIG.  46.    Arts.  157,  158,  159,  160.— 
Adiabatic  Cycles. 

give  the  areas  H  or  h.  Let  us  conceive 
of  a  diagram  in  which  only  one  coor- 
dinate will  at  present  be  named.  That 
coordinate  is  to  be  absolute  temperature. 
Instead  of  specifying  the  other  coordi- 
nate, let  it  be  assumed  that  subtended 
areas  on  this  diagram  are  to  denote 
quantities  of  heat  absorbed  or  emitted, 
just  as  such  areas  on  the  PV  diagram 
represent  external  work  done.  As  an 
example  of  such  a  diagram,  consider 
Fig.  47.  Let  the  substance  be  one 
77 


78  APPLIED  THERMODYNAMICS 

pound  of  water,  initially  at  a  temperature  of  32°  F.,  or  491.6°  abso- 
lute, represented  by  the  height  ab,  the  horizontal  location  of  the 
state  b  being  taken  at  random.  Now  assume  the  water  to  be  heated 
to  212°  F.,  or  671.6°  absolute,  the  specific  heat  being  taken  as  con- 
stant and  equal  to  unity.  The  heat  gained  is  180  B.  t.  u.  The 
final  temperature  of  the  water  fixes  the  vertical  location  of  the 
new  state  point  c?,  i.e.  the  length  of  the  line  cd.  Its  horizontal  lo- 
cation is  fixed  by  the  consideration  that  the  area  subtended  between 
the  path  bd  and  the  axis  which  we  have  marked  ON  shall  be 
180  B.  t.  u.  The  horizontal  distance  ac  may  be  computed  from  the 
properties  of  the  trapezoid  abdc  to  be  equal  to  the  area  abdc  divided 
by  [(«&  +  cd)  +  2]  or  to  180  -r-  [(491.6  +  671.6)  -f-  2]  =  0.31.  The 
point  d  is  thus  located  (Art.  163). 

159.  Application  to  a  Carnot  Cycle.  Ordinates  being  absolute 
temperatures,  and  areas  subtended  being  quantities  of  heat  absorbed 
or  emitted,  we  may  conclude  that  an  isothermal  must  be  a  straight 
horizontal  line;  its  temperature  is  constant,  and  a  finite  amount  of 
heat  is  transferred.  If  the  path  is  from  left  to  right,  heat  is  to  be 

conceived  as  absorbed;  if  from  right  to 
left,  heat  is  rejected.  Along  a  (re- 
versible) adiabatic,  no  movement  of  heat 
occurs.  The  only  line  on  this  diagram 
which  does  not  subtend  a  finite  area  is 
a  straight  vertical  line.  Adiabatics  are 

1       consequently  vertical  stra'ght  lines.     (But 

see  Art.   176.)      The  temperature  must 
N     constantly   change    along   an   adiabatic. 


FIG.  48.   Arts.  159,  160,  161,  163,    The  lengths  of  all  isothermals  drawn  be- 

tween  the  same  two  adiabatics  are  equal. 

The  Carnot  cycle  on  this  new  diagram 
may  then  be  represented  as  a  rectangle  enclosed  by  vertical  and  hori- 
zontal lines ;  and  in  Fig.  48  we  have  a  new  illustration  of  the  cycles 
shown  in  Fig.  46,  all  of  the  lines  corresponding. 

160.    Physical  Significance.     The  new  diagram  is  to  be  conceived 
as  so  related  to  the  P  V  diagram  of  Fig.  46  that  while  an  imaginary 


ENTROPY 


79 


pencil  is  describing  any  stated  path  on  the  latter,  a  corresponding 
pencil  is  tracing  its  path  on  the  former.  In  the  P  V  diagram,  the 
subtended  areas  constantly  represent  external  work  done  by  or  on  the 
substance ;  in  the  new  diagram  they  represent  quantities  of  heat  ab- 
sorbed or  rejected.  (Note,  however,  Art.  176.)  The  area  of  the 
closed  cycle  in  the  first  case  represents  the  net  quantity  of  work  done; 
in  the  second,  it  represents  the  net  amount  of  heat  lost,  and  conse- 
quently, also,  the  net  work  done.  But  subtended  areas  under  a  single 
path  on  the  PV  diagram  do  not  represent  heat  quantities,  nor  in  the 
new  diagram  do  they  represent  work  quantities.  The  validity  of  the 
diagram  is  conditioned  upon  the  absoluteness  of  the  properties  chosen  as 
coordinates.  We  have  seen  that  temperature  is  a  cardinal  property, 
irrespective  of  the  previous  history  of  the  substance  ;  and  it  will  be 
shown  that  this  is  true  also  of  the  horizontal  coordinate,  so  that  we 
may  legitimately  employ  a  diagram  in  which  these  two  properties 
are  the  coordinates. 


161.  Polytropic  Paths.  For  any  path  in  which  the  specific  heat 
is  zero,  the  transfer  of  heat  is  zero,  and  the  path  on  this  diagram  is 
consequently  vertical,  an  adiabatic.  For  specific  heat  equal  to 
infinity,  the  temperature 
cannot  change,  and  the 
path  is  horizontal,  an  iso- 
thermal. For  any  positive 
value  of  the  specific  heat, 
heat  area  and  temperature 
will  be  gained  or  lost 
simultaneously;  the  path 
will  be  similar  to  ai  or  aj, 
Fig.  48.  If  the  specific 
heat  is  negative,  the  tem- 
perature will  increase  with 
rejection  of  heat,  or  de- 
crease with  its  absorption,  as  along  the  paths  ak,  al,  Fig.  48.  These 
results  may  be  compared  with  those  of  Art.  115.  Figure  49  shows 
on  the  new  diagram  the  paths  corresponding  with  those  of  Fig.  31. 
It  may  be  noted  that,  in  general,  though  not  invariably,  increases  of 


FIG.  49.    Arts.  161,  H>5.  —  Polytropic  Paths  on 
Entropy  Diagram. 


80 


APPLIED  THERMODYNAMICS 


volume  are  associated  with  increases  of  the  horizontal  coordinate  of 
the  new  diagram. 

162.  Justification  of  the  Diagram.  In  the  PV  diagram  of  Fig.  50,  consider 
the  cycle  ABCD.  Let  the  heat  absorbed  along  a  portion  of  this  cycle  be  repre- 
sented by  the  infinitesimal  strips  nabN, 
NbcM,  Mcdm,  formed  by  the  indefinitely 
projected  adiabatics.  In  any  one  of  these 
strips,  as  nabN,  we  have,  in  finite  terms, 


nabN_T 
negN'7' 


nabN  _  neaN 
T         ~T~ 


FIG.  50.    Art.  162.  — Entropy  a  Cardinal 
Property. 


Considering  the  whole  series  of  strips 
from  A  to  C,  we  have 

nabN  _  ^-\  negN  « 

or,  using  the  symbol  H  for  heat  trans- 
ferred, 


in  which  T  expresses  temperature  generally. 

Let  the  substance  complete  the  cycle  ABCDA;  we  then  have,  the  paths  being 
reversible, 


while  for  the  cycle  ADCDA, 


whence, 


J<IH_    (      dH        I 
-w- I*  -f+  Y 
ijA  tjc 

n°       nc 

I      4H__    I      dH 

J:  r'J: r  I 


dH 


The  integral 


thus  has  the  same  value  whether  the  path  is  ADC  or  ABC, 


or,  indeed,  any  reversible  path  between  A  and  C  ;  its  value  is  independent  of  the 
path  of  the  substance.  Now  this  integral  will  be  shown  immediately  to  be  the  most 
general  expression  for  the  horizontal  coordinate  of  the  diagram  under  discussion. 
Since  it  denotes  a  cardinal  property,  like  pressure  or  temperature,  it  is  permissible 

to  use  a  diagram  in  which  the  coordinates  are  T  and  f-^r- 


ENTROPY 


81 


163.  Analytical  Expression.  Along  any  path  of  constant  tem- 
perature, as  ab,  Fig.  48,  the  horizontal  distance  nN  may  be  computed 
from  the  expression,  nN=  H  -5-  T,  in  which  H  represents  the  quan- 
tity of  heat  absorbed,  and  T  the  temperature  of  the  isothermal.  If 
the  temperature  varies,  the  horizontal  component  of  the  path  during 
the  absorption  of  dH  units  of  heat  is  dn  =  dH-t-  T.  For  any  path 
along  which  the  specific  heat  is  constant,  and  equals  k,  kdT  '=  dH, 


dn  =         ,  and  *  =  k  =  k  log, 

If  the  specific  heat  is  variable,  say  k  =  a  +  IT,  then 


The  line  Id  of  Fig.  47  is  then  a  logarithmic  curve,  not  a  straight 
line  ;  and  the  method  of  finding  it  adopted  in  Art.  158  is  strictly 
accurate  only  for  an  infinitesimal  change  of  temperature.  Writing 
the  expression  just  derived  in  the  form  n  =  kloge(T-t-  1)  and  remem- 
bering that  T==  PV-±-  R,  while  t  =  pv  -5-  R,  we  have 


The  expression  k\oge(T-r-  f)  is  the  one  most 
commonly  used  for  calculating  values  of  the  hori- 
zontal coordinate  for  polytropic  paths.  The 
expression  dn  =  dH+-  T  is  general  for  all  re- 
versible paths  and  is  regarded  by  Rankine  as 
the  fundamental  equation  of  thermodynamics. 


dn 


FIG.  51.  Art.  164.—  Graphi- 
cal Determination  of 
Specific  Heat. 


164.    Computation  of    Specific  Heat.      If    at    any 

point  on  a  reversible  path  a   tangent   be   drawn,  the 

length  of  the  subtangent  on  the  JV-axis  represents  the 
value  of  the  specific  heat  at 

that  point.  In  Fig.  51,  draw  the  tangent  nm  to  the 
curve  AB  at  the  point  nand  construct  the  infinitesimal 
triangle  dtdn.  From  similar  triangles,  mr:nr::dn:  dt, 
or  mr  =  Tdn  ~  dt  =  dH  -H  dt  =  k  (Art.  58). 

165.   Comparison  of  Specific  Heats.     If  a  gas  is 
heated  at  constant  pressure  from  a,  Fig.  52,  it  will 
N     gain  heat  and  temperature,  following  some  such 

F,G.  4    Art.'lM.-Co.n-      Path    aS   ab'        If      hfiated      a*      C°nStant     ^T6' 
parison  of  Specific  Heats,     through   an   equal  range  of  temperature,   a   i 


82  APPLIED  THERMODYNAMICS 

quantity  of  heat  will  be  gained ;  i.e.  the  subtended  area  aefd  will  be  less 
than  the  area  abed.  In  general,  the  less  the  specific  heat,  the  more 
nearly  vertical  will  be  the  path.  (Compare  Fig.  49.)  When  k  =  0,  the 
path  is  vertical ;  when  k  =  co,  the  path  is  horizontal. 

166.  Properties  of  the  Carnot  Cycle.     In  Fig.  48,  it  is  easy  to  see  that 
since  efficiency  is  equal  to  net  expenditure  of  heat  divided  by  gross  ex- 
penditure, the  ratio  of  the  areas  abdc  and  abNn  expresses  the  efficiency, 
and  that  this  ratio  is  equal  to  (T—t)-t-  T.     The  cycle  abdc  is  obviously 
the  most  efficient  of  all  that  can  be  inscribed  between  the  limiting  iso- 
thermals  and  adiabatics. 

167.  Other  Deductions.     The  net  enclosed  area  on  the   TN  diagram 
represents  the  net  movement  of  heat.     That  this  area  is  always  equivalent 
to  the  corresponding  enclosed  area  on  the  PV  diagram  is  a  statement  of 
the  first  law  of  thermodynamics.     Two  statements  of  the  second  law  have 
just  been  derived  (Art.  166).     The  theorem  of  Art.  106,  relating  to  the 
representation  of  heat  absorbed  by  the  area  between  the  adiabatics,  is 
accepted  upon  inspection  of  the  TN  diagram.    That  of  Art.  150,  from  which 
the  Kelvin  absolute  scale  of  temperature  was  deduced,  is  equally  obvious. 

168.  Entropy.     The  horizontal  or  N  coordinate  on  the  diagram 
now  presented  was  called  by  Clausius  the  entropy  of  the  body.     It- 
may  be  mathematically  defined  as  the  ratio  n  =  \—^  -     The  physical 

definition  or  conception  should  be  framed  by  each  reader  for  himself. 
Wood  calls  entropy  "  that  property  of  the  substance  which  remains 
constant  throughout  the  changes  represented  by  a  [reversible]  adia- 
batic  line."  It  is  for  this  reason  that  reversible  adiabatics  are  called 
isentropics,  and  that  we  have  used  the  letters  n,  N  in  denoting 
adiabatics. 

*\<b 

/ 

169.  General  Formulas.  It  must  be  thoroughly 
understood  that  the  validity  of  the  entropy  diagram  is 
dependent  upon  the  fact  that  entropy  is  a  cardinal  prop- 
erty, like  pressure,  volume,  and  temperature.  For  this 
reason  it  is  desirable  to  become  familiar  with  compu- 
tations of  change  of  entropy  irrespective  of  the  path 
pursued.  Otherwise,  the  method  of  Art.  163  is  usually 

FIG.  54.    Art.  169.— Change    more  convenient. 

of  Entropy.  Consider  the  states  a  and  b,  Fig.  54.     Let  the 

substance  pass  at  constant  pressure  to  c  and  thence  at  constant  volume 


ENTROPY  83 

*•/, 

to  b.     The  entropy  increases  by  k  logc  Z  +1  loge  ^  (Art.  163),  k  and  J 

4m  Tc 

in  this  instance  denoting  the  respective   special  values  of  the  specific 
heats.     An  equivalent  expression  arises  from  Charles'  law  : 

n  =  k  logc  I?  +  i  loge  5  =  k  loge  ^+  *  logc  £,  (A) 

in  which  last  the  final  and  initial  states  only  are  included. 
We  may  also  write, 

Zs 

n  =  I  log,  -JL  +    /V/  °Se  Vf,  Arts.  94,  95,  163, 

-*  a  -*• 

=  J  logp  ^  +  (fc  -  r>  logc  -p,  Arts.  51,  65  :  (B) 

•*•  a  '  n 

and  further, 


The  entropy  is  completely  determined  by  the  adiabatic  through  the  state  point. 

T 

In  the  expression  nl  =  kl  loge—  ,  the  value  of  n{  apparently  depends  upon  that  of  klt 

which  is  of  course  related  to  the  path  ;  along  another  path,  the  gain  or  loss  of 

T 

entropy  might  be  n2  =  k-2  logp—  >  a  different  value;  but  although  the  temperatures 

would  be  the  same  at  the  beginning  and  end  of  both  processes,  the  pressures  or 
volumes  would  differ.  The  states  would  consequently  be  different,  and  the  values 
of  n  should  therefore  differ  also. 

A  graphical  method  for  the  transfer  of  perfect  gas  paths  from  the  PFto  the 
TN  plane  has  been  developed  by  Berry  (1). 

170.  Other  Names  for  n.     Kankine  called  n  the  thermodynamic  func- 
tion.    It  has  been  called  the  "  heat  factor."     Zeuner  describes  it  as  "  heat 
weight."     It  has  also  been  called  the  "mass"  of  heat.     The  letters  T,  N, 
which  we  have  used  in  marking  the  coordinates,  were  formerly  replaced 
by  the  Greek  letters  theta  and  phi,  indicating  absolute  temperatures  and 
entropies  ;  whence  the  name,  theta-phi  diagram.     The  TN  diagram  is  now 
commonly  called  the   temperature-entropy   diagram,  or,  more   briefly,  the 
entropy  diagram. 

171.  Entropy  Units.     Thus  far,  entropy  has  been  considered  as  a  hori- 
zontal distance  on  the  diagram,  without  reference  to  any  particular  zero 
point.     Its  units  are  B.  t.  u.  per  degree  of  absolute  temperature.     Changes 


84  APPLIED  THERMODYNAMICS 

of  entropy  are  alone  of  physical  significance.  The  choice  of  a  zero  point 
may  be  made  at  random  ;  there  is  no  logical  zero  of  entropy.  A  conven- 
ient starting  point  is  to  take  the  adiabatic  of  the  substance  through  the 
state  P  =  2116.8,  T=  32°  F.,  as  the  OTaxis,  the  entropy  of  this  adiabatic 
being  assumed  to  be  zero,  as  in  ordinary  tables.  Thus,  in  Fig.  47  (Art. 
158),  the  0 Taxis  should  be  shifted  to  pass  through  the  point  &,  which 
was  located  at  random  horizontally. 

172.  Hydraulic  Analogy.     The  analogy  of  Art.  140  may  be  extended  to  illus- 
trate the  conception  of  entropy.      Suppose  a  certain  weight  of  water  W  to  be 
maintained  at  a  height  H  above  sea  level ;  and  that  in  passing  through  a  motor 
its  level   is  reduced  to  h.     The   initial  potential   energy  of  the  water  may  be 
regarded  as    WH\   the  final  residual  energy  as  Wh;   the  energy  expended   as 
W(H  —  h).     Let  this  operation  be  compared  with  that  of  a  Carnot  cycle,  taking 
in  heat  at  T  and  discharging  it  at  t.     Regarding  heat  as  the  product  of  N  and  T, 
then  the  heat  energies  corresponding  to  the  water  energies  just  described  are  NT, 
Nt,  and  N(  T  —  t)  ;  N  being  analagous  to  W,  the  weight  of  the  water. 

173.  Adiabatic  Equation.     Consider  the  states  1  and  2,  on  an  adiabatic 
path,  Fig.  55.     The  change  of  entropy  along  the  constant  volume  path  13 

P  rp 

is  I  loge  — 3 ;  that  along  the  constant  pressure  path  32 
TI 

ijt 

is  k   loge  —-     The  difference  of   entropy   between 

*f 
1  and  2,  irrespective  of  the  path,  is 

'   I  log.  £  +  ft  log,  £  =  I  log,  £•  +  ft  loge  I?. 

^1  *3  *i  "\ 


FIG.  55.     Art.  173.—     ^or  a  reversible  adiabatic  process,  this  is  equal  to 
Adiabatic  Equation.         zero  ;   whence 

I  loge  5  =  -  k  log,  %,  or  y  loge  V2  +  loge  P,  =  y  loge  Vt  +  log,  P1? 
*i  *i 

from  which  we  readily  derive  PiF/  =  P.Ff,  the  equation  of  the  adiabatic. 

174.  Use  of  the  Entropy  Diagram.  The  intelligent  use  of  the  entropy 
diagram  is  of  fundamental  importance  in  simplifying  thermodynamic  con- 
ceptions. The  mathematical  processes  formerly  adhered  to  in  presenting 
the  subject  have  been  largely  abandoned  in  recent  text-books,  largely  on 
account  of  the  simplicity  of  illustration  made  possible  by  employing  the 
TN  coordinates. 

Belpaire  was  probably  the  first  to  appreciate  their  usefulness.  Gibbs,  at  about 
the  same  date,  1873,  presented  the  method  in  this  country  and  first  employed  as 
coordinates  the  three  properties  volume,  entropy,  and  internal  energy.  Linde, 


IRREVERSIBLE   PROCESSES 


85 


Schroeter,  Hermann,  Zeuner,  and  Gray  used  TN  diagrams  prior  to  1890.  Cotterill, 
Ayrton  and  Perry,  Dwelshauvers  Dery  and  Ewing  have  employed  them  to  a  con- 
siderable extent.  Detailed  treatments  of  this  thermodynamic  method  have  been 
given  by  Boulvin,  Reeve,  Berry,  and  Golding  (2).  Some  precautions  necessary  in 
its  practical  application  are  suggested  in  Arts.  454-458. 


IRREVERSIBLE  PROCESSES 

175.  Modification  of  the  Entropy  Conception.  It  is  of  importance  to  distinguish 
between  reversible  and  irreversible  processes  in  relation  to  entropy  changes. 
The  significance  of  the  term  reversible,  as  ap- 
plied to  a  path,  was  discussed  in  Art.  125.  A 
process  is  reversible  only  when  it  consists  of  a 
series  of  successive  states  of  thermal  equilib- 
rium. A  series  of  paths  constitute  a  reversible 
process  only  when  they  form  a  closed  cycle, 
each  path  of  which  is  itself  reversible.  The 
Carnot  cycle  is  a  perfect  example  of  a  reversible 
process.  As  an  example  of  an  irreversible  cycle, 
let  the  substance,  after  isothermal  expansion, 
as  in  the  Carnot  cycle,  be  transferred  directly 
to  the  condenser.  Heat  will  be  abstracted,  and 
the  pressure  may  be  reduced  at  constant  vol- 
ume, as  along  be,  Fig.  56.  Then  allow  it  to  compress  isothermally,  as  in  the 
Carnot  cycle,  and  finally  to  be  transferred  to  the  source,  where  the  temperature 
and  pressure  increase  at  constant  volume,  as  along  da.  This  cycle  cannot  be 
operated  in  the  reverse  order,  for  the  pressure  and  temperature  cannot  be  reduced 
from  a  to  d  while  the  substance  is  in  communication  with  the  source,  nor  increased 
from  c  to  b  while  it  is  in  communication  with  the  condenser. 


FIG.  56. 


Art.  175.  — Irreversible 
Cycle. 


176.  Irreversibility  in  the  Porous  Plug  Experiment.  We  have  seen  that  in  this 
instance  of  unresisted  expansion,  the  fundamental  formula  of  Art.  12  becomes 
H  =T  +  I  +  W  +  V  (Art.  127).  Knowing  H  =  0,  W  =  0,  we  may  write 
( T  +  I)  =  —  V,  or  velocity  is  attained  at  the  expense  of  the  internal  energy.  The 
velocity  evidences  kinetic  energy  ;  mechanical  work  is  made  possible  ;  and  we  might 
expect  an  exhibition  of  such  work  and  a  fall  of  internal  energy,  and  consequently 
of  temperature.  But  we  find  no  such  utilization  of  the  kinetic  energy  of  the  rapidly 
flowing  jet ;  on  the  contrary,  the  gas  is  gradually  brought  to  rest  and  the  velocity 
derived  from  an  expenditure  of  internal  energy  is  reconverted  to  internal  energy. 
The  process  was  adiabatic,  for  no  transfer  of  heat  occurred ;  it  was  at  the  same 
time  isothermal,  for  no  change  of  temperature  occurred ;  and  while  both  adiabatic 
and  isothermal,  no  external  work  was  done,  so  that  the  PV  diagram  is  invalid. 

Further :  the  adiabatic  path  here  considered  was  not  isentropic,  like  an  ordinary 
adiabatic.  The  area  under  the  path  on  the  TN  diagram  no  longer  represents  heat 

absorbed  from  surrounding  bodies.     Neither  does  dn  =  -,ior  H  is  zero,  while 


86  APPLIED   THERMODYNAMICS 

n  is  finite.     The  expression  for  increase  of  entropy,   (l      ,  along  a  reversible  path,  does 
not  hold  for  irreversible  operations. 

In  irreversible  operations,  the  expression   f  —  •  ceases  to  represent  a  cardinal 

property.    .We  have  the  following  propositions  : 

(a)  In  a  reversible  operation,  the  sum  of  the  entropies  of  the  participating  substances 
is  unchanged.  During  a  reversible  change,  the  temperatures  of  the  heat-absorbing 
and  heat-emitting  bodies  must  differ  to  an  infinitesimal  extent  only;  they  are  in 
finite  terms  equal.  The  heat  lost  by  the  one  body  is  equal  to  the  heat  gained  by 

the  other,  so  that  the  expression  f-^  denotes  both  the  loss  of  entropy  by  the  one 

substance  and  the  gain  by  the  other;  the  total  stock  of  entropy  remaining  constant. 
(ft)  Du'-ing  irreversible  operations,  the  aggregate  entropy  increases.  Consider  two 
engines  working  in  the  Carnot  cycle,  the  first  taking  the  quantity  of  heat  Hl  from 
the  source,  and  discharging  the  quantity  H2  to  the  condenser;  the  second,  acting 
as  a  heat  pump  (Art.  139),  taking  the  quantity  H2'  from  the  condenser  and  restoring 
//j'  to  the  source.  Then  if  the  work  produced  by  the  engine  is  expended  in  driving 
the  pump,  without  loss  by  friction, 

H^-H2  =  H,>  -  H2'. 

If  the  engine  is  irreversible,  //,>  ///,  or  Hl  -  ///>0,  whence,  H2  -  7/2'>0.     If 
we  denote  by  a   a  positive  finite   value,   Hl  =  ///  +  a  and  H2  =  H2  -f  a.     But 

-f±i  =  7j±,  or  -£  -  -£-  =  0,  and  consequently 
n<i       22          Jl        i2 

Hl  -  a      //a  -  a  ,  77,      H, 

- 


Since  7\  >  Tv  ^  -  ^2  <  0,  or  ^2>^,  or,  generally,    f  ^  <  0.      The  value  of 

ll        J2  J2     .  Jl  J     L 

(  —  -  is  thus,  for  irreversible  operations,  negative. 

Now  let  a  substance  work  irreversibly  from  A  to  fi,  thence  recersibly  from  B  to 
A.     We  may  write 

CB(IH-L   CA(IH       CB<IH       CBdH  ^A 

)      ~T+   I      ~T=    I      ~T~          ^r<°*  (A) 

JA     l       JB     l       JA     1       JA     1 

(irrev.)          (rev.)  (irrev.)  (rev.) 

But  the  change  of  entropy  of  the  substance  in  passing  from  A  to  B  is  NB—  N~A  =  \      -—  » 

JA     T 

dH  being  the  amount  of  heat  absorbed  along  any  reversible  path,  while  the  change 
of   entropy   of   the   source    ichich   supplies   the  substance   with   heat    (reversibly)    is 

NBf  -  NA'  =  -  I     '—  ,  the  negative  sign  denoting  that  heat  has  been  abstracted. 

JA     1 
We  have  then,  from  equation  (A), 

-  (AY  -  NA')  -  (N£  -  NA~)  <  0;  or,  (JV/,  +  NB')  -  (NA  +  NA')>0: 

i.e.  the  sum  of  the  entropies  of  the  participating  substances  increases  when  the 
process  is  irreversible. 


'  IRREVERSIBILITY  87 

(c)  The  loss  of  work  due  to  irreversibility  is  proportional  to  the  increase  of  entropy. 
Consider  a  partially  completed  cycle :  one  which  might  be  made  complete  were  all 
of  the  paths  reversible.  Let  the  heat  absorbed  be  Q,  at  the  temperature  T,  in- 
creasing the  entropy  of  the  substance  by  ^,;  and  let  its  entropy  be  further  increased 
by  N1  —  N  during  the  process.  The  total  increase  of  entropy  is  then  n  =  N'  —  N+  — , 

whence  Q  =  T(n  —  N'  +  N).  The  work  done  may  be  written  as  H  —  H'  +  Q,  in 
which  H  and  //'  are  the  initial  and  final  heat  contents  respectively.  Calling  this 

W,  we  have 

W  =  11  -  H'  +  T(n  -  N'  +  N). 

/•     J  TT 

In  a  reversible  cycle  J  —  =  n  -  0 ;  whence  WR  =  H  -  If  +  T(N  -  N')  and 
WR  -  W=  Tn. 

(A  careful  distinction  should  be  made  at  this  point  between  the  expression 

and  the  term  entropy.     The  former  is  merely  an  expression  for  the  latter 

under  specific  conditions.  In  the  typical  irreversible  process  furnished  by  the 
porous  plug  experiment,  the  entropy  increased;  and  this  is  the  case  generally  with 

such  processes,  in  which  </n>—  •     Internal  transfers  of  heat  may  augment  the 

entropy  even  of  a  heat-insulated  body,  if  it  be  not  in  uniform  thermal  condition. 
Perhaps  the  most  general  statement  possible  for  the  second  law  of  thermody- 
namics is  that  all  actual  processes  tend  to  increase  the  entropy ;  as  we  have  seen,  this 
keeps  possible  efficiencies  below  those  of  the  perfect  reversible  engine.  The  prod- 
uct of  the  increase  of  entropy  by  the  temperature  is  a  measure  of  the  waste  of 
energy  (3).) 

Most  operations  in  power  machinery  may  without  serious  error  be  analyzed 
as  if  reversible ;  unrestricted  expansions  must  always  be  excepted.  The  entropy 
diagram  to  this  extent  ceases  to  have  "an  automatic  meaning." 

(1)  Tlie  Temperature-Entropy  Diagram,  1908.  (2)  See  Berry,  op.  cit.  (3)  The 
works  of  Preston,  Swinburne,  and  Planck  may  be  consulted  by  those  interested  in  this 
aspect  of  the  subject. 


SYNOPSIS  OF  CHAPTER  VIII 

It  is  impracticable  to  measure  PFheat  areas  between  the  adiabatics. 

The  rectangular  diagram  :  ordinates  =  temperature  ;  areas  =  heat  transfers. 

Application  to  a  Carnot  cycle  :  a  rectangle. 

The   validity  of  the  diagram  is  conditioned  upon  the  absoluteness  of  the  horizontal 

coordinate. 
The  slope  of  a  path  of  constant  specific  heat  depends  upon  ihevahte  of  the  specific  heat. 

The   expression    (^  has  a  definite  value  for  any  reversible   change  of  condition, 

J    T 
regardless  of  the  path  pursued  to  effect  the  change. 

dn  =  <UI,  or  n  =  k  logre  -  for  constant  specific  heat  =  k,  or  n  =  a  loge  ^+  b(T-  0  for 
variable  specific  heat  —  a  -f  b  T. 


88  APPLIED  THERMODYNAMICS 

The  value  of  the  specific  heat  along  a  poly  tropic  is  represented  by  the  length  of  the  sub- 

tangent. 
Illustrations  :  comparison  of  k  and  I  ;  efficiency  of  Carnot  cycle  ;  the  first  law  ;  the 

second  law  ;  heat  area  between  adiabatics  ;  Kelvin's  absolute  scale. 
Entropy  units  are  B.  t.  u.per  degree  absolute.     The  adiabatic  for  zero  entropy  is  at 
P=  2116.8. 


=  jfe  !Oge    ±  +  (jfc  _  ?)lOgc          - 
«  -i  a  -/a  V  a  -La  -*& 

Hydraulic  analogy  ;  physical  significance  of  entropy  ;  use  of  the  diagram. 
Derivation  of  the  adiabatic  equation. 

Irreversible  Processes 

A  reversible  cycle  is  composed  of  reversible  paths  ;  example  of  an  irreversible  cycle. 

Joule's  experiment  as  an  example  of  irreversible  operation. 

Heat  generated  by  mechanical  friction  of  particles  ;  the  path  both  isothermal  and  adia- 

batic, but  not  isentropic. 
H=  T+I+  W+  For   r  =  -(/+  T). 


For  irreversible  processes,  dn  is  not  equal  to  —  ;  the  subtended  area  does  not  repre- 

sent a  transfer  of  heat  ;  non-isentropic  adiabatics. 

In  reversible  operations,   the  aggregate  entropy  of  the  participating  substances  is 
unchanged. 

During  irreversible  operations,  the  aggregate  entropy  increases,  and  (*^<0. 

The  loss  of  work  due  to  increase  of  entropy  is  nT;  dn>  dH 

T 


PROBLEMS 

1.  Plot  to  scale  the  TWpath  of  one  pound  of  air  heated  (a)  at  constant  pressure 
from  100°  F.  to  200°  F.,  then  (6)  at  constant  volume  to  300°  F.     The  logarithmic 
curves  may  be  treated  as  two  straight  lines. 

2.  Construct  the  entropy  diagram  for  a  Carnot  cycle  for  one  pound  of  air  in  which 
T=  400°  F.,  t  =  100°  F.,  and  the  volume  in  the  first  stage  increases  from  1  to  4  cubic 
feet.    Do  not  use  the  formulas  in  Art.  169. 

3.  Plot  on  the  TN  diagram  paths  along  which  the  specific  heats  are  respectively 
0,  QO,  3.4,  0.23,  0.17,  -0.3,   -10.4,  between  T  =  459.6  and.  T=  919.2,  treating  the 
logarithmic  curves  as  straight  lines. 

4.  The  variable  specific  heat  being  0.20-0.0004  T-  0.000002  T2  (T  being  in 
Fahrenheit  degrees),  plot  the  TN  path  from  100°  F.  to  140°  F.  in  four  steps,  using 
mean  values  for  the  specific  heat  in  each  step. 

Find  by  integration  the  change  of  entropy  during  the  whole  operation. 

5.  What  is  the   specific  heat   at  T  =  40  (absolute)  for  a  path  the  equation  of 
which  on  the  TN  diagram  is  TN  =  c  =  1200  ? 

6.  Confirm  Art.  134  by  computation  from  the  TN  diagram. 

7.  Plot  the  path  along  which  1  unit  of  entropy  is  gained  per  100°  absolute.     What 
is  the  mean  specific  heat  along  this  path  from  0°  to  400°  absolute  ? 


ENTROPY  89 

8.  What  is  the  entropy  measured  above  the  arbitrary  zero  per  pound  of  air  at 
normal  atmospheric  pressure  in  a  room  at  70°  F.  ? 

9.  Find  the  arbitrary  entropy  of  a  pound  of  air  in  the  cylinder  of  a  compressor  at 
2000  Ib.  pressure  per  square  inch  and  142°  F. 

10.  Find  the  entropy  of  a  sphere  of  hydrogen  10  miles  in  diameter  at  atmospheric 
pressure  and  175°  F. 

11.  The  specific  heat  being  0.24  +  0.0002  T,  find  the  increase  in  entropy  between 
459.6  and  919.2  degrees,  all  absolute.     What  is  the  mean  specific  heat  over  this  range  ? 


CHAPTER   IX 

COMPRESSED  AIR 

177.  Compressed  Air  Engines.     Engines  are  sometimes  used  in  which  the 
working  substance  is  cold  air  at  high  pressure.     The  pressure  is  previously  pro- 
duced by  a  separate  device ;  the  air  is  then  transmitted  to  the  engine,  the  latter 
being  occasionally  in  the  form  of  an  ordinary  steam  engine.     This  type  of  motor 
is  often  used  in  mines,  on  locomotives,  or  elsewhere  where  excessive  losses  by  con- 
densation would  follow  the  use  of  steam.     For  small  powers,  a  simple  form  of 
rotary  engine  is  sometimes  employed,  on  account  of  its  convenience,  and  in  spite 
of  its  low  efficiency.     The  absence  of  heat,  leakage,  danger,  noise,  and  odor  makes 
these  motors  popular  in  those  cities  where  the  public  distribution  of  compressed 
air  from  central  stations  is  practiced  (1).     The  exhausted  air  aids  in  ventilating 
the  rooms  in  which  it  is  used. 

178.  Other  Uses  of  Compressed  Air.     Aside  from  the  driving  of  engines,  high- 
pressure  air  is  used  for  a  variety  of  purposes  in  mines,  quarries,  and  manufac- 
turing plants,  for  operating  hoists,  forging  and  bending  machines,  punches,  etc., 
for  cleaning  buildings,  for  operating  "  steam  "  hammers,  and  for  pumping  water 
by  the  ingenious  "air  lift"  system.     In  many  works,  the  amount  of  power  trans- 
mitted by  this  medium  exceeds  that  conveyed  by  belt  and  shaft  or  electric  wire. 
The  air  is  usually  compressed  by  steam  power,  and  it  is  obvious  that  a  loss  must 
occur  in  the  transformation.     This  loss  may  be  offset  by  the  convenience  and  ease 
of  transmitting  air  as  compared  with  steam ;  the  economical  generation,  distribu- 
tion, and  utilization  of  this  form  of  power  have  become  matters  of  first  importance. 

The  first  applications  were  made  during  the  building  of  the  Mont  Cenis  tun- 
nel through  the  Alps,  about  1860  (2).  Air  was  there  employed  for  operating 
locomotives  and  rock  drills,  following  Colladon's  mathematical  computation  of 
the  small  loss  of  pressure  during  comparatively  long  transmissions.  A  general 
introduction  in  mining  operations  followed.  Two-stage  compressors  with  inter- 
coolers  were  in  use  in  this  country  as  early  as  1881.  Among  the  projects  sub- 
mitted to  the  international  commission  for  the  utilization  of  the  power  of  Niagara, 
there  were  three  in  which  distribution  by  compressed  air  was  contemplated.  Wide- 
spread industrial  applications  of  this  medium  have  accompanied  the  perfecting  of 
the  small  modern  interchangeable  "pneumatic  tools." 

179.  Air  Machines  in  General.    In  the  type  of  machinery  under  consideration, 
a  considerable  elevation  of  pressure  is  attained.     Centrifugal  fans  or  paddle-wheel 
blowers,  commonly  employed  in  ventilating  plants,  move  large  volumes  of  air  at 
very  slight  pressures,  —  usually  a  fraction  of  a  pound,  —  and  the  thermodynamic 

90 


THE  AIR   ENGINE 


91 


relations  are  unimportant.  Rotary  blowers  are  used  for  moderate  pressures,  —  up 
to  20  lb.,  —  but  they  are  generally  wasteful  of  power  and  are  principally  employed 
to  furnish  blast  for  foundry  cupolas,  forges,  etc.  The  machine  used  for  com- 
pressing air  for  power  purposes  is  ordinarily  a  piston  compressor,  mechanically 
quite  similar  to  a  reciprocating  steam  engine.  These  compressors  are  sometimes 
employed  for  comparatively  low  pressures  also,  as  "blowing  engines." 


THE  -AiR  ENGINE 

180.  Air  Engine  Cycle.     In  Fig.  57,  ABCD  represents  an  ideal- 
ized air  engine  cycle.     AB  shows  the  admission  of  air  to  the  cylin- 
der.    Since  the  latter  is  small  as  compared  with  the  transmitting 
pipe  line,  the  specific  volume  and  pres- 
sure of   the    fluid,    and    consequently 

its  temperature  as  well,  remain  un- 
changed. BC  represents  expansion 
after  the  supply  from  the  mains  is 
cut  off.  If  the  temperature  at  B  is  that 
of  the  external  atmosphere,  and  ex- 
pansion proceeds  slowly,  so  that  any 
fall  of  temperature  along  BO  is  offset 
by  the  transmission  of  heat  from  the 
-outside  air  through  the  cylinder  walls, 
this  line  is  an  isothermal.  If,  however, 
expansion  is  rapid,  so  that  no  transfer 

of  heat  occurs,  BO  will  be  an  adiabatic.  In  practice,  the  expansion 
line  is  a  polytropic,  lying  usually  between  the  adiabatic  and  the 
isothermal.  CD  represents  the  expulsion  of  the  air  from  the  cyl- 
inder at  the  completion  of  the  working  stroke.  At  J>,  the  inlet 
valve  opens  and  the  pressure  rises  to  that  at  A.  The  volumes 
shown  on  this  diagram  are  not  specific  volumes,  but  volumes  of  air  in 
the  cylinder.  Subtended  areas,  nevertheless,  represent  external  work. 

181.  Modified  Cycle.     The  additional  work  area  LMC  obtained  by  ex- 
pansion beyond  some  limiting  volume,  say  that  along  xy,  is  small.     A 
slight  gain  iii  efficiency  is  thus  made  at  the  cost  of  a  much  larger  cylin- 
der.    In  practice,  the  cycle  is  usually  terminated  prior  to  complete  expan- 
sion, and  has  the  form  ABLMD,  the  line  LM  representing  the  fall  of 
pressure  which  occurs  when -the  exhaust  valve  opens. 


FIG.  57.    Arts.  180-183,  189,  222,  223, 
226,  Prob.  6.— Air  Engine  Cycles. 


92  APPLIED  THERMODYNAMICS 

182.  Work  Done.  Letting  p  denote  the  pressure  along  AB,  P 
the  pressure  at  the  end  of  the  expansion,  q  the  "  back  pressure  " 
along  MD  (slightly  above  that  of  the  atmosphere),  and  letting  v 
denote  the  volume  at  J5,  and  Fthat  at  the  end  of  expansion,  both 
volumes  being  measured  from  OA  as  a  line  of  zero  volumes,  then, 
for  isothermal  expansion,  the  work  done  is 

y 

pv+pvloge  --  qV\ 

and  for  expansion  such  that  pvn  =  P  F",  it  is 

v-PV 


- 
n—  1 


Tr 
qV. 


183.  Maximum  Work.  Under  the  most  favorable  conditions,  expan- 
sion would  be  isothermal  and  "  complete  "  ;  i.e.  continued  down  to  the 
back-pressure  line  CD.  Then,  q  =  P  =  pv-^-  F,  and  the  work  would  be 
pv  loge(F-7-  v).  For  complete  adiabatic  expansion,  the  work  would  be 


184.  Entropy  Diagram.     This  cannot  be  obtained  by  direct  transfer  from  the 
PV  diagram,  because  we  are  dealing  with  a  varying  quantity  of  air.     The  method 
of  deriving  an  illustrative  entropy  diagram  is  explained  in  Art.  218. 

185.  Fall  of   Temperature.     If  air  is  received  by  an  engine  at 
P,  T,  and  expanded  to  p,  t,  then  from  Art.  104,  if  P  -^p=  10,  and 
T=  529°  absolute,  with  adiabatic  expansion,  t=  —  187°  F. 

This  fall  of  temperature  during  adiabatic  expansion  is  a  serious  matter. 
Low  final  temperatures  are  fatal  to  successful  working  if  the  slightest 
trace  of  moisture  is  present  in  the  air,  on  account  of  the  formation  of  ice 
in  the  exhaust  valves  and  passages.  This  difficulty  is  counteracted  in 
various  ways  :  by  circulating  warm  air  about  the  exhaust  passages  ;  by 
specially  designed  exhaust  ports;  by  a  reduced  range  of  pressures;  by 
avoidance  of  adiabatic  expansion  (Art.  219)  ;  and  by  thoroughly  drying 
the  air.  The  jacketing  of  the  cylinder  with  hot  air  has  been  proposed. 
Unwin  mentions  (3)  the  use  of  a  spray  of  water,  injected  into  the  air 
while  passing  through  a  preheater  (Art.  186).  This  reaches  the  engine 
as  steam  and  condenses  during  expansion,  giving  up  its  latent  heat  of 
vaporization  and  thus  raising  the  temperature.  In  the  experiments  on 
the  use  of  compressed  air  for  street  railway  traction  in  New  York,  stored 
hot  water  was  employed  to  preheat  the  air.  The  only  commercially  sue- 


PREHEATERS 


93 


cessful  method  of  avoiding  inconveniently  low  temperatures  after  expan- 
sion is  by  raising  the  temperature  of  the  inlet  air. 

186.  Preheaters.     In  the  Paris   installation  (4),    small   heaters   were 
placed  at  the  various  engines.     These  were  double  cylindrical  boxes  of 
cast  iron,  with  an  intervening  space  through  which  the  air  passed  in  a 
circuitous  manner.     The  inner  space  contained  a  coke  fire,  from  which 
the  products  of  combustion  passed  over  the  top  and  down  the  outside  of 
the  outer   shell.     For  a  10-hp.  engine,  the  extreme   dimensions   of  the 
heater  were  21  in.  in  diameter  and  33  in.  in  height. 

187.  Economy  of  Preheaters.     The  heat  used  to  produce  elevation  of 
temperature  is  not  wasted.     The  volume  of  the  air  is  increased,  and  the 
weight    consumed    in     the 

engine  is  correspondingly 
decreased.  Kennedy  esti- 
mated in  one  case  that 
the  reduction  in  air  con- 
sumption due  to  the  in- 
crease of  volume  should 
have  been,  theoretically, 
0.30;  actually,  it  was  0.25. 
The  mechanical  efficiency 
(Art.  214)  of  the  engine 
is  improved  by  the  use  of 
preheated  air.  In  ^^^ 
one  instance,  Ken-  |JJ 
nedy  computed  a 
saving  of  225  cu.  ft.  of 
"free"  air  (i.e.  air  at  at- 
mospheric pressure  and  tem- 
perature'} to  have  been  ef- 
fected at  an  expenditure 
of  0.4  Ib.  of  coke.  Unwin 
found  that  all  of  the  air 
used  by  a  72-hp.  engine 
could  be  heated  to  300°  F. 
by  15  Ib.  of  coke  per  hour. 
Figure  58  represents  a 
modern  form  of  preheater. 


FIG.  58.    Art.  187.—  Rand  Air  Preheater. 


188.   Volume  of  Cylinder.      If  n  be  the  number  of  single  strokes  per 
minute  of  a  double-acting  engine,  Fthe  cylinder  volume  (maximum  vol- 


94  APPLIED  THERMODYNAMICS 

ume  of  fluid),  TFthe  number  of  pounds  of  air  used  per  minute,  v  the  specific 
volume  of  the  air  at  its  lowest  pressure  p  and  its  temperature  t,  N  the 
horse  power  of  the  engine,  and  U  the  work  done  in  foot-pounds  per  pound 
of  air,  then,  ignoring  clearance  (the  space  between  the  piston  and  the  cyl- 
inder head  at  the  end  of  the  stroke),  the  volume  swept  through  by  the 

piston  per  minute  =  Wv  =  2  nV=  WR-\ 

whence  F=  -^^j  and  since  WU  =  33,000  N,  W=  380QO  N9 
2np  U 

33000  NRt 


and  V= 


2nUp 


189.  Compressive  Cycle.     For  quiet  running,  as  well  as  for  other 
reasons,  to  be  discussed  later,  it  is  desirable  to  arrange  the  valve 
movements  so  that  some  air  is  gradually  compressed  into  the  clear- 
ance space  during  the  latter  part  of  the  return  stroke,  as  along  Ea, 
Fig.  57.     This  is  accomplished  by  causing  the  exhaust  valve  to  close 
at  E,  the  inlet  valve  opening  at  a.     The  work  expended  in  this  com- 
pression is  partially  recovered  during  the  subsequent  forward  stroke, 

•the  air  in  the  clearance  space  acting  as  an  elastic  cushion. 

190.  Actual  Design.     A  single-acting  10-hp.  air  engine  at  100  r.  p.  m., 
working  between  114.7  and  14.7  Ib.  absolute  pressure,  with  an  "appar- 
ent "  (Art.  450)  volume  ratio  during  expansion  of  5  :  1  and  clearance  equal 
to  5  per  cent  of  the  piston  displacement,  begins  to  compress  when  the 
return  stroke  of  the  piston  is  T%-  completed.     The  expansion  and  compres- 
sion curves  are  PF13  =  c.     Assuming  that  the  actual  engine  will  give  90 
per  cent  of   the  work  theoretically  computed,  find  the  size  of  cylinder 
(diameter  =  stroke)  and  the  free  air  consumption  per  Ihp.-hr. 

In  Fig.  59,  draw  the  lines  ab  and  cd  representing  the  pressure  limits.  We  are 
to  construct  the  ideal  PV  diagram,  making  its  enclosed  length  represent,  to  any 
convenient  scale,  the  displacement  of  the  piston  per  stroke.  The  extreme  length 
of  the  diagram  from  the  oP  axis  will  be  5  per  cent  greater,  on  account  of  clear- 
ance. The  limiting  volume  lines  ef  and  gh  are  thus  sketched  in  ;  BC  is  plotted, 

making  -^-  =  5  ;  the  point  E  is  taken  so  that  S?  =  0.9,  and  EF  drawn.     Then 
AB  Di 

ABCDEF  is  the  ideal  diagram.     We  have,  putting  Di  =  D, 

PA  =  PB  =  114.7. 


VB  =  0.25  D. 

VF  -  VA  =  0.05  D. 

VK  =  0.15  D. 


DESIGN   OF   AIR  ENGINE 


95 


PcVc"=  P.Vf  or  P,  =  P 


"  =  114.7(5||)U  =  17.75. 


Work  per  stroke  =jABi  +  iBCm  -  EDmk  -jFEk 

=  P.(V»-  ro  +  P*F-  f  '*'- 


=  144[(114.7  x  0.20  D)  -f  014.7  x  0.25  Z»^-(  17.75  x  1.05  /)) 


-  air  x  o  o  D\    (6L31  x  °-05  J>  -  a4-7  x  °- 

0.3 


=  5803.2  D  foot-pounds. 

The  actual  engine  will  then  give  0.9  x  5803.2  D  =  5222.88  D  foot-pounds  per  stroke 
or  5222.88  D  x  100  foot-pounds  per  minute,  which  is  to  be  made  equal  to  10  hp.,or 


6         114.7 


C  — 17.75 


k        i 
FIG.  59.     Art.  190. — Design  of  Air  Engine. 

to  10  x  33,000  foot-pounds.  Then  522,288  D  =  330,000  and  D  =  0.63  cu.  ft.  Since 
the  diameter  of  the  engine  equals  its  stroke,  we  write  0.7854  d2  x  d  =  0.63  x  1728, 
where  d  is  the  diameter  in  inches:  whence  d  =  11.15  in. 

To  estimate  the  air  consumption :  at  the  point  B,  the  whole  volume  of  air  is 
0.25  D.  Part  of  this  is  clearance  air,  used  repeatedly,  and  not  chargeable  to  the 
engine.  The  clearance  air  at  E  had  the  volume  VE  and  the  pressure  PE.  If  its 


96 


APPLIED  THERMODYNAMICS 


behavior  conforms  to  the  law  PF1-8  =  c,  then  at  the  pressure  of  114.7  Ib.  (point  G) 
we  would  have  i 

PGVG«  =  PEVE"  or  VG  =  V1'3  =  0-15  #°'7    -  0.0309  D. 


The  volume  of  fresh  air  brought  into  the  cylinder  per  stroke  is  then 
0.25  D  -  0.0309  D  =  0.2191  D 

or,  per  hour,  0.2191  x  0.63  x  100  x  60  =  828  cu.  ft.     Reduced  to  free  air  (Art.  186), 

1147 
this  would  be  828  x  -=±  =  6450  cu.  ft.,  or  645  cu.  ft.  per  Ihp.-hr.     (Compare 


Art.  192.) 


14.7 


191.  Effect  of  Early  Compression.     If  compression  were  to  begin  at  a  suffi- 
ciently early  point,  so  that  the  pressure  were  raised  to  that  in  the  supply  pipe 
before  the  admission  valve  opened,  the  fresh  air  would  find  the  clearance  space 
already  completely  filled,  and  a  less  quantity  of  such  fresh  air,  by  0.05  D,  instead 
of  0.0309  Z>,  would  be  required. 

192.  Actual  Performances  of  Air  Engines.     Kennedy  (5)  found  a  con- 
sumption of  890  cu.  ft.  of  free  air  per  Ihp.-hr.,  in  a  small  horizontal  steam 
engine.     Under  the  conditions  of  Art.  183,  the  theoretical  maximum  work 
which  this  quantity  of  air  could  perform  is  1.27  hp.     The  cylinder  effi- 
ciency (Art.  215)  of  the  engine  was  therefore  1.0-5-1.27=0.79.     With 
small  rotary  engines,  without  expansion,  tests  of  the  Paris  compressed  air 
system  showed  free  air  consumption  rates  of  from  1946  to  2330  cu.  ft. 
By  working  these   motors   expansively,  the  rates  were   brought  within 
the  range  from  848  to  1286  cu.  ft.     A  good  reciprocating  engine  with  pre- 
heated air  realized  a  rate  of  477  cu.  ft.,  corresponding  to  36.4  Ib.,  per 
brake  horse  power  per  hour.     The  cylinder  efficiencies  in  these  examples 
varied  from  0.368  to  0.876,  and  the  mechanical  efficiencies  (Art.  214)  from 
0.85  to  0.92. 

THE  AIR  COMPRESSOR 

193.  Action   of   Piston   Compressor.     Figure    60   represents    the 
parts  concerned  in  the  cycle  of  an  air  compressor.     Air  is  drawn 

from  the  atmosphere  through  the  spring  check 
valve  a,  filling  the  space  C  in  the  cylinder.  This 
inflow  of  air  continues  until  the  piston  has 
reached  its  extreme  right-hand  position.  On  the 
return  stroke,  the  valve  a  being  closed,  compres- 
sion proceeds  until  the  pressure  is  slightly  greater 
than  that  in  the  receiver  D.  The  balanced  outlet 
valve  b  then  opens,  and  air  passes  from  C  to  D 
at  practically  constant  pressure.  When  the  pis- 


FIG.  60.     Art.  193.— 
Piston  Compressor. 


THE  AIR  COMPRESSOR 


97 


ton  reaches  the  end  of  its  stroke,  there  will  still  remain  the  clear- 
ance volume  of  air  in  the  cylinder.  This  expands  during  the  early 
part  of  the  next  stroke  to  the  right,  but  as  soon  as  the  pressure  of 
this  air  falls  slightly  below  that  of  the  atmosphere,  the  valve  a  again 
opens. 


194.  Cycle.  An  actual  diagram  is  given, 
as  ADOS,  Fig.  61.  Slight  fluctuations  in 
pressure  occur  during  discharge  along  AD  and 
during  suction  along  CB-,  the  mean  discharge 
pressure  must  of  course  be  slightly  greater  FlG-  G1-  Art-  1J)*.— Cycle 

of  Air  Compressor. 

than  the  receiver  pressure,  and  the  mean  suc- 
tion pressure  slightly  less  than  atmospheric  pressure.     Eliminating 
these  irregularities  and  the  effect  of  clearance,  the  ideal  diagram 
is  adcb. 


195.  Form  of  Compression  Curve.  The  remarks  in  Art.  180  as  to 
the  conditions  of  isothermal  or  adiabatic  expansion  apply  equally  to  the 
compression  curve  BA.  Close  approximation  to  the  isothermal  path  is  the 

ideal  of  compressor  per- 
formance. Let  A,  Fig.  62, 
be  the  point  at  which 
compression  begins,  and 
let  DE  represent  the 
maximum  pressure  to  be 
attained.  Let  the  cycle 
be  completed  through  the 
states  F,  G.  Then  the 
work  expended,  if  com- 
pression is  isothermal,  is 
ACFG\  if  adiabatic,  the 

WQ]^  expen(le(l  js  ABFG 

The  same  amount  of  air 


FIG.  <j± 


Arts.  195,  197,  213,  218.  — Forms  of  Compression 
Curve. 


has  been  compressed,  and  to  the  same  pressure,  in  either  case;  the  area 
ABC  represents,  therefore,  needlessly  expended  work.  Furthermore,  dur- 
ing transmission  to  the  point  at  which  the  air  is  to  be  applied,  in  the 
great  majority  of  cases,  the  air  will  have  been  cooled  down  practically 
to  the  temperature  of  the  atmosphere  ;  so  that  even  if  compressed  adia- 
batically,  with  rise  of  temperature,  to  B,  it  will  nevertheless  be  at  the 
state  C  when  ready  for  expansion  in  the  consumer's  engine.  If  it  there 


98  APPLIED  THERMODYNAMICS 

again  expand  adiabatically  (along  CH)  instead  of  isothermally  (along 
CA),  a  definite  amount  of  available  power  will  have  been  lost,  repre- 
sented by  the  area  CIIA.  During  compression,  we  aim  to  have  the  work 
area  small;  during  expansion  the  object  is  that  it  be  large;  the  adiabatic 
path  prevents  the  attainment  of  either  of  these  ideals. 

The  loss  of  power  by  adiabatic  compression  is  so  great  that  various 
methods  are  employed  to  produce  an  approximately  isothermal  path.  As 
a  general  rule,  the  path  is  consequently  intermediate  between  the  iso- 
thermal and  the  adiabatic,  a  polytropic,  pvn  =  C.  The  relations  derived 
in  Arts.  183  and  185  for  adiabatic  compression  apply  equally  to  this  path, 
excepting  that  for  y  we  must  write  w,  the  value  of  n  being  somewhere 
between  1.0  and  1.402.  The  effect  of  water  in  the  cylinder,  whether  in- 
troduced as  vapor  with  the  air,  or  purposely  injected,  is  to  somewhat 
reduce  the  value  of  n,  to  increase  the  interchange  of  heat  with  the  walls, 
and  to  cause  the  line  FG,  Fig.  62,  to  be  straight  and  vertical,  rather  than 
an  adiabatic  expansion,  thus  slightly  increasing  the  capacity  of  the  com- 
pressor, as  shown  in  Art.  222. 

196.  Temperature  Rise.     The  rise  of  temperature  due  to  compression  may  be 
computed  as  in  Art.  185.     Under  ordinary  conditions,  the  air  leaves  the  com- 
pressor at  a  temperature  higher   than   that  of  boiling  water.     Without  cooling 
devices,  it  may  leave  at  such  a  temperature  as  to  make  the  pipes  red  hot.     It  is 
easy  to  compute  the  (not  very  extreme)  conditions  under  wKich  the  rise  in  tem- 
perature would  be  sufficient  to  melt  the  cast-iron  compressor  cylinder. 

197.  Computation  of  Loss.     The  uselessly  expended  work  during  adiabatic 
(and  similarly,  during  any  other  than  isothermal)  compression  may  be  directly 
computed  from  the  difference  of  the  work  areas   CAKI  and  CBAKI,  Fig.  62. 
The  work  under  the  isothermal  is  (p,  v,  referring  to  the  point  C,  and  P,  V,  to 
the  point  A),  pvloge  (F-4-  i?)  =  _pyloge  (/>  -4-  P)  ;  while  if  Q  is  the  volume  at  B, 
the  work  under  ABC  is 


_1 

i 

but  pQ»  =  PVy  and  Q  =  T7(—  V; 

\pl 

so  that  the  percentage  of  loss  corresponding  to  any  ratio  of  initial  and  final  pres- 
sures and  any  terminal  (or  initial)  volume  may  be  at  once  computed. 

198.  Basis  of  Methods  for  Improvement.  Any  value  of  n  exceeding  1.0  for 
the  path  of  compression  is  due  to  the  generation  of  heat  as  the  pressure  rises, 
faster  than  the  walls  of  the  cylinder  can  transmit  it  to  the  atmosphere.  The  high 
temperatures  thus  produced  introduce  serious  difficulties  in  lubrication.  Economi- 
cal compression  is  a  matter  of  air  cooling  ;  while,  on  the  consumer's  part,  economy 
depends  upon  air  heating. 


COMPRESSION   CURVE 


99 


199.  Air  Cooling.     In  certain  applications,  where  a  strong  draft  is  available, 
the  movement  of  the  atmosphere  may  be  utilized  to  cool  the  compressor  cylinder 
walls  and  thus  to  chill  the  working  air  during  compression.     While  this  method 
of  cooling  is  quite  inadequate,  it  has  the  advantage  of  simplicity  and  is  largely 
employed  on  the  air  "pumps"  which  operate  the  brakes  of  railway  trains. 

200.  Injection  of  Water.     This  was  the  method  of  cooling  originally  em- 
ployed at  Mont  Cenis  by  Colladon.     Figure  63  shows  the  actual  indicator  card 
(Art.  484)  from  one  of  the  older  Colladon 

compressors.  EBCDisthe  corresponding 
ideal  card  with  isothermal  compression. 
The  cooling  by  stream  injection  was  evi- 
dently not  very  effective.  Figure  64  rep- 
resents another  diagram  from  a  compressor 
in  which  this  method  of  cooling  was  em- 
ployed ;  ab  representing  the  isothermal  and 
ac  the  adiabatic.  The  exponent  of  the 
actual  curve  ad  was  1.36;  the  gain  over 
adiabatic  compression  was  very  slight.  By 
introducing  the 
water  in  a  very 

fine  spray,  a  somewhat  lower  value  of  the  exponent 
was  obtained  in  the  compressors  used  by  Colladon  on 
the  St.  Gothard  tunnel.     Gause  and  Post  (6)  have  re- 
duced the  value  of  n  to  1.26  by  an  atomized  spray. 
Figure  65  shows  one  of  their  diagrams,  ab  being  the 
isothermal  and  ac  the  adiabatic.     In  all  cases, 
spray  injection  is  better  than  solid  stream  in- 
jection.     The   value   n  =  1.36,  above  given, 
was  obtained   when  a  solid  jet  of  half-inch 
diameter  was  used.    It  is  estimated  that  errors 
of    the    indicator  may    introduce   an  uncer- 


FIG.  63. 


Art.  200.  — Card  from  Colladon 
Compressor. 


FIG. 


Art.  200.  —  Cooling  by  Jet 
Injection. 


tainty  amounting  to  0.02  in  the  value  of  n.  Piston  leakage  would  cause  an 
apparently  low  value.  The  comparative 
efficiency  of  spray  injection  is  shown  from 
the  fairly  uniform  temperature  of  dis- 
charged air,  which  can  be  maintained  even 
with  a  varying  speed  of  the  compressor. 
In  the  Gause  and  Post  experiments,  with 
inlet  air  at  81|°  F.,  the  temperature  of  dis- 
charge was  95°  F.  Spray  injection  has  the 
objection  that  it  fills  the  air  with  vapor,  and 
it  has  been  found  that  the  orifices  must  be 
so  small  that  they  soon  clog  and  become  ^IG> 
inoperative.  The  use  of  either  a  spray  or 


Art.  200.— Cooling  by  Atomized 
Spray. 


a  solid  jet  causes  cutting  of  the  cylinder  and  piston  by  the  gritty  substances  carried 
in  the  water.     In  American  practice  the  injection  of  water  has  been  abandoned. 


100 


APPLIED  THERMODYNAMICS 


201 .   Water  Jackets.     These  reduce  the  value  of  n  to  a  very  slight  ex- 
tent only,  but  are  generally  employed  because  of  their  favorable  influence 

on  cylinder  lubrication.  Gause  and 
Post  found  that  with  inlet  air  at 
81°  F.,  and  jackets  on  the  barrels  of 
the  cylinders  only  (not  on  the  heads), 
the  temperature  of  the  discharged  air 
was  320°  F.  Cooling  occurred  dur- 
ing expulsion  rather  than  during  com- 
pression. The  cooling  effect  depends 
largely  upon  the  heat  transmissive 
power  of  the  cylinder  walls,  and  the 
value  of  n  consequently  increases  at 
high  speeds.  Two  specimen  cards 
are  given  in  Fig.  66 ;  ab  being  the  isothermal  and  ac  the  adiabatic.  With 
more  thorough  cooling,  jacketed 
heads,  etc.,  a  lower  value  of  n 
may  be  obtained ;  but  this  value 
is  seldom  or  never  below  1.3. 
Figure  67  shows  a  card  given 
by  Unwin  from  a  Cockerill  com- 
pressor, DC  indicating  the  ideal 
isothermal  curve.  At  the 
higher  pressures,  air  is  appar- 
ently more  readily  cooled ;  its 

own   heat-conducting    power   is        ^^^^^^^^^M  ^^^^^^^-^     D1 
increased. 


FIG.  66.    Art.  201.— Cooling  by  Jackets. 


202.  Heat  Abstracted.  In 
Fig.  68,  let  AB  and  AC  be  the 
adiabatic  and  the  actual  paths, 


o 


E' 

D 

FIG.  67.    Art.  201.  — Cockerill  Compressor  with 
Jacket  Cooling. 


An  and  CN  adiabatics ;  the  heat  to  be  abstracted  is  then  equivalent  to 

NCAn  =  I  ACL  +  nAIE  -  NCLE. 

Now  IACL  = 


n-1  '  y- 

NCLE  = 


whence 


2/-1' 
pv 


n  — 


FIG.  68.     Arts.   202,   203. -Heat  Ab-    This  is  the  heat  to  be   abstracted  per 
stracted  by  Cooling  Agent.  volume  V  at  pressure  P,  compressed  to 


MULTI-STAGE  COMPRESSION 


101 


p,  expressed  in  foot-pounds.  For  isothermal  compression,  as  along 
AD,  IACL=pv  loge  (FH-  v),  and  the  total  heat  to  be  abstracted  is  measured 
by  this  area.  If  the  path  is  adiabatic,  AB,  n  =  y,  and  the  expression  for 
heat  abstraction  becomes  zero.* 

203.    Elimination  of  v.     It  is  convenient  to  express  the  total  area  NCAn  in 
terms  of  p,  P,  and  V  only.     The  area 

=i 


PV  (  po     n    pvr(p\*=i     -] 
=  ^fp-1)=—  [Llpj 


Also, 


y-l 

.hence  NCAn  =  ±t  f  (*)  v'  -  l]  + 
n—  ILVP/  J 


y-\p 


204.  Water  Required.     Let  the  heat  to  be  abstracted,  as  above  com- 
puted, be  H,  in  heat  units.     Then  if  S  and  s  are   the  final  and  initial 
temperatures  of  cooling  water,  and  C  the  weight  of  water  circulated,  we 
have  C=H-s-(S  —  s),  the  specific  heat  of  water  being  taken  as  1.0.     In 
practice,  the  range  of  temperature  of  the  cooling  water  may  be  from  40° 
to  70°  F. 

205.  Multi-stage  Compression.     The  effective  method  of  securing  a 
low  value  of  n  is  by  multi-stage  operation,  the  principle  of  which  is 
illustrated  in  Fig.  69.     Let  A  be  the 

state  at  the  beginning  of  compres- 
sion, and  let  it  be  assumed  that  the 
path  is  practically  adiabatic,  in  spite 
of  jacket  cooling,  as  AB.  Let  AC 
be  an  isothermal.  In  multi-stage 
compression,  the  air  follows  the  path 
AB  up  to  a  moderate  pressure,  as  at 
D,  and  is  then  discharged  and  cooled 
at  constant  pressure  in  an  external 
vessel,  until  its  temperature  is  as 
nearly  as  possible  that  at  which  it  was  admitted  to  the  cylinder. 
The  path  representing  this  cooling  is  DE.  The  air  now  passes  to 

*  More  simply,  as  suggested  by  Chevalier,  the  specific  heat  along  AC  is  s  =  I -- 
(Art.  112):   the  heat  to  be  abstracted  is  then 


Art.   205.  —  Multi-stage  Com- 
pression. 


102 


APPLIED  THERMODYNAMICS 


FIG.  70.    Arts.  205,  206.  —  Two-stage  Com- 
»  .'jjressor  Indicator  Diagram. 


a  second  cylinder,  is  adiabatically  compressed  along  EF,  ejected  and 
cooled  along  FGr,  and  finally  compressed  in  still  another  cylinder 

along  G-H.  The  diagram  illus- 
trates compression  in  three 
"  stages  "  ;  but  two  or  four  stages 
are  sometimes  used.  The  work 
saved  over  that  of  single  stage 
adiabatic  compression  is  shown 
by  the  irregular  shaded  area 
HG-FEDB,  equivalent  to  a  re- 
duction in  the  value  of  n,  under 
good  conditions,  from  1.402  to 
about  1.25.  Figure  70  shows  the  diagram  from  a  two-stage  2000  hp. 
compressor,  in  which  solid  water  jets  were  used  in  the  cylinders. 
The  cooling  water  was  at  a  lower 
temperature  than  the  inlet  air, 
causing  the  point  h  to  fall  inside 
the  isothermal  curve  AB.  The 
compression  curves  in  each  cyl- 
inder give  71  =  1.36.  Figure  71 
is  the  diagram  for  a  two-stage 
Riedler  compressor  with  spray  in- 
jection, AB  being  an  isothermal 
and  AC  an  adiabatic. 


FIG.  71. 


Arts.  205,  214.— Two-stage  Riedler 
Compressor  Diagram. 


206.  Intercooling.     Some  work  is  always  wasted  on  account  of  the  friction  of 
the  air  passing  through  the  intercooling  device.     In  early  compressors,  this  loss 
often  more  than  outweighed  the  gain  due  to  compounding.     The  area  ghij,  Fig. 
70,  indicates  the  work  wasted  from  this  cause.     In  this  particular  instance,  the 
loss  is  exceptionally  small.     Besides  this,  the  additional  air  friction  through  two 
or  more  sets  of  valves  and  ports,  and  the  extra  mechanical  friction  due  to  a  multi- 
plication of  cylinders  and  reciprocating  parts  must  be  considered.     Multi-stage 
compression  does  not  pay  unless  the  intercooling  is  thoroughly  effective.     It  seldom 
pays  when  the  pressure  attained  is  low.     Incidental  advantages  in  multi-stage 
operation  arise  from  reduced  mechanical  strains  (Art.  462),  higher  volumetric 
efficiency  (Art.  226),  better  lubrication,  and  the  removal  of  moisture  by  precipita- 
tion during  the  intercooling. 

207.  Types  of  Intercoolers.     The  "external  vessel"  of  Art.  205  is  called  the 
intercooler.     It  consists  usually  of  a  riveted  or  cast-iron  cylindrical  shell,  with  cast- 


INTERCOOLING 


103 


iron  heads.     Inside  are  straight  tubes  of  brass  or  wrought  iron,  running  between 
steel  tube  sheets.     The  back  tube  sheet  is  often  attached  to  a  stiff  cast-iron  inter- 


FIG.  72.    Art.  207.  —  Allis-Chalmers  Horizontal  Intercooler. 

nal  head,  so  that  the  tubes,  sheet,  and  head 
are  free  to  move  as  the  tubes  expand 
(Fig.  72).  The  air  entering  the  shell  sur- 
rounds the  tubes  and  is  compelled  by  baffles 
to  cross  the  tube  space  on  its  way  to  the  out- 
let. Any  moisture  precipitated  is  drained 
off  at  the  pipe  a.  The  water  is  guided  to 
the  tubes  by  internally  projecting  ribs  on 
the  heads,  which  cause  it  to  circulate  from 
end  to  end  of  the  intercooler,  several  times. 
If  of  ample  volume,  as  it  should  be,  the 
intercocler  serves  as  a  receiver  or  storage 
tank.  The  one  illustrated  is  mounted  in 
a  horizontal  position.  A  vertical  type  is 
shown  in  Fig.  73.  The  funnel  provides  a 
method  of  ascertaining  at  all  times  whether 
water  is  flowing. 

.  i 

208.  Aftercoolers.  In  most 
manufacturing  plants,  the  pres- 
ence of  moisture  in  the  air  is  ob- 
jectionable, on  account  of  the 
difficulty  of  lubrication  of  air 
tools,  and  because  of  the  rapid  de- 
struction of  the  rubber  hose  used 
for  connecting  these  tools  with 
the  pipe  line.  To  remove  the 
moisture  (and  some  of  the  oil)  FIG.  73.  Art.  207.— Ingersoll-Sergeaut  Vertical 
present  after  the  last  stage  of  com-  Intercooler. 


104  APPLIED  THERMODYNAMICS 

pression,  and  by  cooling  the  air  to  decrease  the  necessary  size  of  transmitting  pipe, 
aftercoolers  are  employed.  They  are  similar  in  design  and  appearance  to  inter- 
coolers.  An  incidental  advantage  arising  from  their  use  is  the  decreased  strain 
on  the  pipe  line  following  the  introduction  of  air  at  a  more  nearly  normal  tem- 
perature. 

209.   Power   Consumed.     From    Art.  98,   the   work   under   any  curve 
pva=PV»is,  adopting  the  notation  of  Art.  202,      n~ 


pv  J      n-l(         \p4 

The  work  along  an  adiabatic  is  expressed  by  the  last  formula  if  we  make 
n  =  y  =  1.402.  The  work  of  expelling  the  air  from  the  cylinder  after  com- 
pression is  pv.  The  work  of  drawing  the  air  into  the  cylinder,  neglecting 

.- j 

clearance,  is  PV=pv(  —  )      -     The  net  work  expended  in  the  cycle  is  the 

\PJ 
algebraic  sum  of  these  three  quantities,  which  we  may  write, 


p 

It  is  usually  more  convenient  to  eliminate  v,  the  volume  after  compres 
sion.     This  gives  the  work  expression, 


If  pressures  are  in  pounds  per  square  inch,  the  foot-pounds  of  work  per 
minute  will  be  obtained  by  multiplying  this  expression  by  the  number  of 
working  strokes  per  minute  and  by  144  ;  and  the  theoretical  horse  power 
necessary  for  compression  may  be  found  by  dividing  this  product  by 
33,000.  If  we  make  "F=l,  P=14.7,  we  obtain  the  power  necessary  to 
compress  one  cubic  foot  of  free  air.  If  the  air  is  to  be  used  to  drive  a 
motor,  it  will  then  in  most  cases  have  cooled  to  its  initial  temperature 
(Art.  195),  so  that  its  volume  after  compression  and  cooling  will  be 
PV-*-p.  The  work  expended  per  cubic  foot  of  this  compressed  and 
cooled  air  is  then  obtained  by  multiplying  the  work  per  cubic  foot  of  free 

air  by  £• 

210.  Work  of  Compression.  In  some  text-books,  the  work  area  under  the 
compression  curve  is  specifically  referred  to  as  the  work  of  compression.  This  is 
not  the  total  work  area  of  the  cycle. 


RECEIVER  PRESSURE  105 

211.  Range  of  Stages  in  Multi-stage  Compression.  Let  the  lowest  pres- 
sure be  g,  the  highest  p,  and  the  pressure  during  intercooliug  P.  Also  let 
intercooling  be  complete,  so  that  the  air  is  reduced  to  its  initial  tempera- 
ture, so  that  the  volume  V  after  intercooling  is  ^,  in  which  r  is  the 
volume  at  the  beginning  of  compression  in  the  first  cylinder.  Adopting 

the  second  of  the  work  expressions  just  found,  and  writing  z  for  n  ~~    ,  we 
have  n 

Work  in  first  stage  =  ^  j  f- Y  -  1 1 . 
Work  in  second  stage  =  ^!(|Y- 1}  =^  {(|Y-  1 

Total  work  =  2T  {  fP\M+fP\*  _  2 1  =  W. 
Differentiating  with  respect  to  P,  we  obtain 


_ 


dP 

ri/pv-1 

—  nr)-l  —  1       — 

P2VP 


i/py-^ 

9\9J 


For  a  minimum  value  of  W,  the  result  desired  in  proportioning  the  pres- 
sure ranges,   this  expression  is  put  equal  to  zero,  giving 

P-=pq,  or  P  =  ^/pq. 

An  extension  of  the  analysis  serves  to  establish  a  division  of  pressures 
for  four  stage  machines.  From  the  pressure  ranges  given,  it  may  easily 
be  shown  that  in  the  ideal  cycle  the  condition  of  minimum  work  is  that  the 
amounts  of  work  done  in  each  of  the  cylinders  be  equal.  The  number  of 
stages  increases  as  the  range  of  pressures  increases ;  in  ordinary  practice, 
the  two-stage  compressor  is  employed,  with  final  pressures  of  about  100 
Ib.  per  square  inch  above  the  pressure  of  the  atmosphere. 

/ 
ENGINE  AND  COMPRESSOR  RELATIONS 

212.    Losses  in  Compressed  Air  Systems.     Starting  with   mechanical  power 
delivered  to  the  compressor,  we  have  the  following  losses:  — 
(a)  friction    of    the    compressor    mechanism,    affecting    the    mechanical 
efficiency; 


106  APPLIED  THERMODYNAMICS 

(6)  thermodynamic  loss,  chiefly  from  failure  to  realize  isothermal  com- 
pression, but  also  from  friction  and  leakage  of  air,  clearance,  etc., 
indicated  by  the  cylinder  efficiency ; 

(c)  transmissive  losses  in  pipe  lines  ; 

(d)  thermodynamic  losses  at  the  consumer's  engine,  like  those  of  (b) ; 

(e)  friction  losses  at  the  consumer's  engine,  like  those  of  (a). 

213.  Compressive  Efficiency.     While  not  an  efficiency  in  the  true  sense  of  the 
term,  the  relation  of  work  generated  during  expansion  in  the  engine  to  that  ex- 
pended during  compression  in  the  compressor  is  sometimes  called  the  compressive 
efficiency.     It  is  the  quotient  of  the  areas  FCHG  and  FBA  G,  Fig.  62.     From  the 
expression  in  Art.  209  for  work  under  a  polytropic  plus  work  of  discharge  along 
BF  or  of  admission  along  FC,  we  note  that,  the  values  of  P  and  p  being  identical 
for  the  two  paths,  AB  and  CH,  in  question,  the  total  work  under  either  of  these 
paths  is  a  direct  function  of  the  volume  V  at  the  lower  pressure  P.     In  this  case, 
providing  the  value  of  n  be  the  same  for  both  paths,  the  two  work  areas  have  the 
ratio  V  -+-  x,  where  V  is  the  volume  at  A,  and  x  that  at  H.     It  follows  that  all  the 
ratios  of  volumes  LN  -4-  LM,  OQ  -f-  OP,  etc.,  are  equal,  and  equal  to  the  ratio  of 

areas.     The  compressive  efficiency,  then,  —  —  =  T  +-  t,  where  t  is  the  temperature 

at  A  (or  that  at  C),  and  T  that  at  H.  For  isothermal  paths,  T  =  t,  and  the  com- 
pressive efficiency  is  unity.  In  various  tests,  the  compressive  efficiency  has  ranged 
from  0.488  to  0.898.  It  depends,  of  course,  on  the  value  of  n;  increasing  as  n  de 
creases. 

214.  Mechanical  Efficiency.     For  the  compressor,  this  is  the  quotient  of  work 
expended  in  the  cylinder  by  work  consumed  at  the  fly  wheel ;  for  the  engine,  it 
is  the  quotient  of  work  delivered  at  the  fly  wheel  by  work  done  in  the  cylinder. 

Friction  losses  in  the  mechanism  measure  the  mechanical  inefficiency  of  the 
compressor  or  engine.  With  no  friction,  all  of  the  power  delivered  would  be  ex- 
pended in  compression,  and  all  of  the  elastic  force  of  the  air  would  be  available 
for  doing  work,  and  the  mechanical  efficiency  would  be  1.0.  In  practice,  since 
compressors  are  usually  directly  driven  from  steam  engines,  with  piston  rods  in 
common,  it  is  impossible  to  distinguish  between  the  mechanical  efficiency  of  the 
compressor  and  that  of  the  steam  engine.  The  combined  efficiency,  in  one  of  the 
best  recorded  tests,  is  given  as  0.92.  For  the  compressor  whose  card  is  shown  in 
Fig.  71,  the  combined  efficiency  was  0.87.  Kennedy  reports  an  average  figure  of 
0.845  (7).  Unwin  states  that  the  usual  value  is  from  0.85  to  0.87  (8).  These 
efficiencies  are  of  course  determined  by  comparing  the  areas  of  the  steam  and  air 
indicator  cards. 

215.  Cylinder  Efficiency.     The  true  efficiency,  thermodynamically  speaking, 
is  indicated  by  the  ratio  of  areas  of  the  actual  and  ideal  PV  diagrams.     For  the 
compressor,  the  cylinder  efficiency  is  the  ratio  of  the  work  done  in  the  ideal  cycle, 
without  clearance,  drawing  in  air  at  atmospheric  pressure,  compressing  it  isothermally 

'  and  discharging  it  at  the  constant  receiver  pressure,  to  the  work  done  in  the  actual  cycle 
of  the  same  maximum  volume.  It  measures  item  (6)  (Art.  212).  It  is  not  the  "com- 


PLANT   EFFICIENCY  107 

pressive  efficiency  "  of  Art.  213.  For  the  engine,  it  is  the  ratio  of  the  work  done  in 
the  actual  cycle  to  the  work  of  an  ideal  cycle  without  clearance,  tvith  isothermal  expan- 
sion to  the  same  maximum  volume  from  the  same  initial  state,  and  with  constant  pressures 
during  reception  and  discharge  ;  the  former  being  that  of  the  pipe  line  and  the  latter  that 
oflhe  atmosphere.  Its  value  may  range  from  0.70  to  0.90  in  good  machines,  in  gen- 
eral increasing  as  the  value  of  n  decreases.  An  additional  influence  is  fluid  fric- 
tion, causing,  in  the  compressor,  a  fall  of  pressure  through  the  suction  stroke  and 
a  rise  of  pressure  during  the  expulsion  stroke  ;  a:id  in  the  engine,  a  fall  of  pressure 
during  admission  and  excessive  back  pressure  during  exhaust.  All  of  these  condi- 
tions alter  the  area  of  the  PV  cycle.  In  well-designed  machines,  these  losses 
should  be  small.  A  large  capacity  loss  in  the  cylinder  is  still  to  be  considered. 

216.  Discussion  of  Efficiencies.     Considering  the  various  items  of  loss  sug- 
gested in  Art.  212,  we  find  as  average  values  under  good  conditions, 

(a)  mechanical  efficiency,  0.85  to  0.90;  say  0.85; 

(5)  cylinder  efficiency  of  compressor,  0.70  to  0.90;  say  0.80; 

(<?)  transmission  losses,  as  yet  undetermined  ; 

(d)  cylinder  efficiency  of  air  engine,  0.70  to  90.0;  say  0.70; 

(e)  mechanical  efficiency  of  engine,  0.80  to  0.90;  say  0.80. 

The  combined  efficiency  from  steam  cylinder  to  work  performed  at  the  con- 
sumer's engine,  assuming  no  loss  by  transmission,  would  then  be,  as  an  average, 

0.85  x  0.80  x  0.70  x  0.80  =  0.3808. 

For  the  Paris  transmission  system,  Kennedy  found  the  over-all  efficiency  (includ- 
ing pipe  line  losses,  4  per  cent)  to  be  0.26  with  cold  air  or  0.384  with  preheated 
air,  allowing  for  the  fuel  consumption  in  the  preheaters  (9). 

217.  Maximum  Efficiency.     In  the  processes  described,  the  ideal  efficiency  in 
each  case  is  unity.    We  are  here  dealing  not  with  thermodynamic  transformations 
between  heat  and  mechanical  energy,  but  only  with  transformations  from  one  form 
of  mechanical  energy  to  another,  in  part  influenced  by  heat  agencies.     No  strictly 
thermodynamic  transformation  can  have  an  efficiency  of  unity,  on  account  of  the 
limitation  of  the  second  law. 

218.  Entropy  Diagram.     Figure  62  may  serve  to  represent  the  com- 
bined ideal  PV  diagrams  of  the  compressor  (GABF)  and  engine  (FCHG). 


- 

The  quotient  -         -  is  the  compressive  efficiency.     The  area  representing 
GABF 

net  expenditure  of  work  is  CBAH,  bounded  ideally  by  two  adiabatics  or  in 
practice  by  two  poly  tropics  (not  ordinarily  isodiabatics)  and  two  paths  of 
constant  pressure.  This  area  is  now  to  be  illustrated  on  the  TN  coordi- 
nates. 


108 


APPLIED  THERMODYNAMICS 


FIG.  74.    Art. 


For  convenience,  we  reproduce  the  essential  features  of  Fig.  62 
in  Fig.  74.     In  Fig.  75,  lay  off  the  isothermal   T,  and  choose  the 

point  A  at  random.  Now 
if  either  TB  or  TH  be 
given,  we  may  complete 
the  diagram.  Assume 
that  the  former  is  given ; 
then  plot  the  correspond- 
ing isothermal  in  Fig.  75. 
Draw  AB,  an  adiabatic, 
BC  and  AH  as  lines 
of  constant  pressure 

218. —Engine  and  Compressor  Diagrams,    f  n  =  k  \Oge  —  J,    the    point 

C  falling  on  the  isothermal  T.     Then  draw  CH,  an  adiabatic,  de- 

T       T 

termining  the  point  H-,  or,  from  Art.  213,  noting  that  — -  =  -^-,  we 

may  find  the  point  H  di- 
rectly. If  the  paths  AB 
and  CH  are  not  adia- 
batics,  we  may  compute 
the  value  of  the  specific 
heat  from  that  of  n  and 
plot  these  paths  on  Fig. 
75  as  logarithmic  curves  ; 
but  if  the  values  of  n  are 
different  for  the  two 
paths,  it  no  longer  holds 

,,     ,    TB  __  TA        T,  FIG.  75.    Arts.  218,  219,  221.— Compressed  Air  System, 

Hr~~T"  Entropy  Diagram. 

CBAH  in  Fig.  75   now   represents   the   net   work    expenditure   in 
heat  units. 


219.  Comments.  As  the  exponents  of  the  paths  AB  and  CH  decrease, 
these  paths  swerve  into  new  positions,  as  AE,  CD,  decreasing  the  area 
representing  work  expenditure.  Finally,  with  n  =  1,  isothermal  paths, 
the  area  of  the  diagram  becomes  zero ;  a  straight  line,  CA.  Theoretically, 
with  water  colder  than  the  air,  it  might  be  possible  to  reduce  the  tempera- 


ENTROPY  DIAGRAMS 


109 


ture  of  the  air  during  compression,  giving  such  a  cycle  as  AICDA,  or  even, 
with  isothermal  expansion  in  the  engine,  AICA;  in  either  case,  the  net 
work  expenditure  might  be  nega- 
tive; the  cooling  water  accomplish- 
ing the  result  desired. 

220.  Actual  Conditions.  Under 
the  more  usual  condition  that  the 
temperature  of  the  air  at  admission 
to  the  engine  is  somewhat  higher 
than  that  at  which  it  is  received  by 
the  compressor,  we  obtain  Figs. 
76,  77.  T,  Tc  and  either  TB  or  TH 
must  now  be  given.  The  cycle  in 
which  the  temperature  is  reduced 
during  compression  now  appears  FIG.  75. 
as  AICDA  or  AIJA. 


Art.  220. — Usual  Combination  of 
Diagrams. 


FIG.  77.    Art.  220.  —  Combined  Entropy  Diagrams 


221.  Multi-stage  Operation.  Let  the  ideal  pv  path  be  DECS  A,  Fig.  78. 
The  "  triangle "  ABC  of  Fig.  75  is  then  replaced  by  the  area  DECBA, 
Fig.  79,  bounded  by  lines  of  constant  pressure  and  adiabatics. 


The  area 


FIG.  78.    Art.  221.  —  Three-stage  Com- 
pression and  Expansion. 


FIG.  79.    Art.  221.  — Entropy  Diagram, 
Multi-stage  Compression. 


110 


APPLIED  THERMODYNAMICS 


saved  is  BFEC,  which  approaches  zero  as  the  pressure  along  CE,  Fig.  78, 
approaches  that  along  AB  or  at  D,  and  becomes  a  maximum  at  an  inter- 
mediate position,  already  determined  in 
Art.  211.  With  inadequate  intercooling, 
the  area  representing  work  saved  would  be 
yFEx.  Figures  80  and  81  represent  the 
ideal  pv  and  nt  diagrams  respectively  for 
compressor  and  engine,  each  three-stage, 
with  perfect  intercooling  and  aftercooling 
and  preheating  and  with  no  drop  of  pres- 
sure in  transmission.  BbA  and  AliB 
would  be  the  diagrams  with  single-stage 
adiabatic  compression  and  expansion. 

6 


FIG.    80.     Art.    221.  —  Three-stage 
Compression  and  Expansion. 


FIG.  81.    Art.  221.  — Three-stage  Compression  and  Expansion. 

COMPRESSOR  CAPACITY 

222.  Effect  of  Clearance  on  Capacity.  Let  A  BCD,  Fig.  57,  be  the  ideal  pv  dia- 
gram of  a  compressor  without  clearance.  If  there  is  clearance,  the  diagram  will 
be  aBCE;  the  air  left  in  the  cylinder  at  a  will  expand,  nearly  adiabatically,  along 
aE,  so  that  its  volume  at  the  intake  pressure  will  be  somewhat  like  DE.  The 
total  volume  of  fresh  air  taken  into  the  cylinder  cannot  be  DC  as  if  there  were  no 
clearance,  but  is  only  EC.  The  ratio  EC  :  DC  is  called  the  volumetric  efficiency. 
It  is  the  ratio  of  free  air  drawn  in  to  piston  displacement. 

223.  Volumetric  Efficiency.     This  term  is  sometimes  incorrectly  applied  to  the 
factor  1  —  c,  in  which  c  is  the  clearance,  expressed  as  a  fraction  of  the  cylinder 
volume.     This  is  illogical,  because  this  fraction  measures  the  ratio  of  clearance  air 
at  final  pressure,  to  inlet  air  at  atmospheric  pressure  (Aa  -+-  DC,  Fig.  57)  ;  while 
the  reduction  of  compressor  capacity  is  determined  by  the  volume  of  clearance  air 
at  atmospheric  pressure.     If  the  clearance  is  3  per  cent,  the  volumetric  efficiency  is 
much  less  than  97  per  cent. 

224.  Friction  and  Compressor  Capacity.     If  the  intake  ports  or  pipes  are  small, 
an  excessive  suction  will  be  necessary  to  draw  in  the  charge,  and  the  cylinder  will 


VOLUMETRIC   EFFICIENCY 


111 


be  filled  with  air  at  less  than  atmospheric  pressure.  Its  equivalent  volume  at 
atmospheric  pressure  will  then  be  less  than  that  of  the  cylinder.  This  is  shown 
in  Fig.  82.  The  line  of  atmospheric  pressure  is  DF,  the  capacity  is 
reduced  by  FG,  and  the  volumetric  efficiency  is  DF  -f-  HG.  The  capacity 
may  be  seriously  affected  from  this  cause,  in  the  case  of  a  badly  designed 
machine. 


i 


FIG.  82.    Art.  224.  — Effect  of  Suction  Friction. 


225.   Volumetric  Efficiency  ;  Other  Factors.    Where  jackets  or  water  jets 

are  used,  the  air  is  often 
somewhat  heated  during 
the  intake  stroke,  increas- 
ing its  volume,  and  thus, 
as  in  Art.  224,  lowering 
the  volumetric  efficiency. 
The  effect  is  more  notice- 
able with  jacket  cooling, 
with  which  the  cylinder 
walls  often  remain  con- 
stantly at  a  temperature  above  that  of  boiling  water.  Tests  have  shown  a  loss 
of  capacity  of  5  per  cent,  due  to  changing  from  spray  injection  to  jacketing.  —  A 
high  altitude  for  the  compressor  results  in  its  being  supplied  with  rarefied  air,  and 
this  decreases  the  volumetric  efficiency  as  based  on  air  under  standard  pressure. 
At  an  elevation  of  10,000  ft.  the  capacity  falls  off  30  per  cent.  This  is  sometimes 
a  matter  of  importance  in  mining  applications.  —  Volumetric  efficiency,  in  good 
designs,  is  principally  a  matter  of  low  clearance.  The  clearance  of  a  cylinder  is 
practically  constant,  regardless  of  its  length ;  so  that  its  percentage  is  less  in  the 
case  of  the  longer  stroke  compressors.  Such  compressors  are  comparatively 
expensive.  —  When  water  is  injected  into  the  cylinder,  as  is  often  the  case  in 
European  practice,  the  clearance  space  may  be  practically  filled  with  water  at  the 
end  of  the  discharge  stroke.  Water  does  not  appreciably  expand  as  the  pressure 
is  lowered;  so  that  in  these  cases  the  volumetric  efficiency  may  be  determined  by 
the  expression  1  —  c  of  Art.  223,  being  much  greater  than  in  cases  where  water 
injection  is  not  practiced. 


226.  Volumetric  Efficiency  in  Multi-stage  Compression.     Since  the  effect 
of  multi-stage  compression  is  to  reduce  the  pressure  range,  the  expansion 
of  the  air  caught  in  the  clearance  space  is  less,  and  the   distance   DE, 
Fig.  57,  is  reduced.     This  makes   the   volumetric  efficiency,   EG  H-  DC, 
greater  than  in  single  stage  cylinders.     If  FGH  represent  the  line  of  in- 
termediate pressure,  the  ratio  JE  -*-  DC  is  the  gain  in  volumetric  efficiency. 

227.  Refrigeration  of  Entering  Air.     Many  of  the  advantages  following  multi- 
stage operation  and  intercooling  have  been  otherwise  successfully  realized  by  the 
plan  of  cooling  the  air  drawn  into  the  compressor.     This  of  course  increases  the 
density  of  the  air  at  atmospheric  pressure,  and  greatly  increases  the  volumetric 
efficiency.     Incidentally,  much  of  the  moisture  is  precipitated.     At  the  Isabella 


112 


APPLIED  THERMODYNAMICS 


furnace  of  the  Carnegie  Steel  Company,  at  Etna,  Pennsylvania,  a  plant  of  this 
kind  has  been  installed.  An  ordinary  ammonia  refrigerating  machine  cools  the 
air  from  80°  to  28°  F.  This  should  decrease  the  specific  volume  in  the  ratio 
(459.6  +  28)  H-  (459.6  +  80)  =  0.90.  The  free  air  capacity  should  consequently 
be  increased  in  about  this  ratio  (10). 

228.  Typical  Values.     Excluding  the  effect  of  clearance,  a  loss  in  ca- 
pacity of  from  6  to  22  per  cent  has  been  found  by  Unwin  (11).  to  be  due 
to  air  friction  losses  and  to  heating  of  the  entering  air.     Heilemann  (12) 
finds  volumetric  efficiencies  from  0.73  to  0.919.     The  volumetric  efficiency 
could  be  precisely  determined  only  by  measuring  the  air  drawn  in  and 
discharged. 

229.  Volumetric  and  Thermodynamic  Efficiencies.     The  volumetric  effi- 
ciency is  a  measure  of  the  capacity  only.     It  is  not  an  efficiency  in  the  sense 
of  a  ratio  of  "  effect "  to  "  cause."     In  Fig.  83  the  solid  lines  show  an  actual 
compressor  diagram,  the  dotted  lines,  EGHB,  the  corresponding  perfect 
diagram,  with  clearance  and  isothermal  compression.     In  the  actual  case 
we  have  the  wasted  work  areas, 

HLJQ,  due  to  friction  in  discharge  ports ; 
GQKDt  due  to  non-isothermal  compression; 
DFMC,  due  to  friction  during  the  suction  of  the  air. 

At  BHC,  there  is  an  area  representing,  apparently,  a  saving  in  work 
expenditure,  due  to  the  expansion  of  the  clearance  air ;  this  saving  in 

work  has  been  accomplished,  however, 
with  a  decreased  capacity  in  the  pro- 
portion BC  -r-  BE,  a  proportion  which 
is  greater  than  that  of  BHC  to  the  total 
work  area.  Further,  expansion  of  the 
clearance  air  is  made  possible  as  a  result 
of  its  previous  compression  along  FDK\ 
and  the  energy  given  up  by  expansion 
can  never  quite  equal  that  expended  in 

compression.      The   effect   of   excessive 
FIG.    83.      Art.    229.  —  Volumetric    and      -  .     .  ,      .  ,      . 

Thermodynamic  Efficiencies.  friction    during    suction,    reducing    the 

capacity    in    the     ratio    DE  -*-  BE,   is 

usually  more  marked  on  the  capacity  than  on  the  work.  Both  suction 
friction  and  clearance  decrease  the  cylinder  efficiency  as  well  as  the 
volumetric  efficiency,  but  the  former  cannot  be  expressed  in  terms  of 
the  latter.  In  fact,  a  low  volumetric,  efficiency  may  decrease  the  work 
expenditure  absplutely,  though  not  relatively.  An  instance  of  this  is  found 
in  the  case  of  a  compressor  working  at  high  altitude.  Friction  during  dis- 


COMPRESSOR   DESIGN  113 

charge  decreases  the  cylinder  efficiency  (note  the  wasted  work  area  HLJQ\ 
but  is  practically  without  effect  on  the  capacity. 

COMPRESSOR  DESIGN 

230.  Capacity.  The  necessary  size  of  cylinder  is  calculated  much  as  in 
Art.  190.  Let  p,  v,  t,  be  the  pressure,  volume,  and  temperature  of  dis- 
charged air  (v  meaning  the  volume  of  air  handled  per  minute),  and  P,  F,  T, 
those  of  the  inlet  air.  Then,  since  PF-f-  T  =  pv  -T-  t,  the  volume  drawn 
into  the  compressor  per  minute  is  V=pvT  -s-  Pt,  provided  that  the  air  is 
dry  at  both  intake  and  delivery.  If  n  is  the  number  of  revolutions  per 
minute,  and  the  compressor  is  double-acting,  then,  neglecting  clearance, 

the  piston  displacement  per  stroke  is  V  -*-  2  n  =  f^-  —  . 


This  computation  of  capacity  takes  no  account  of  volumetric  losses. 
In  some  cases,  a  rough  approximation  is  made,  as  described,  and  by 
slightly  varying  the  speed  of  the  compressor  its  capacity  is  made  equal  to 
that  required.  Allowance  for  clearance  may  readily  be  made.  Let  the 
suction  pressure  be  P,  the  final  pressure  p,  the  clearance  volume  at  the 

final  pressure  —  of  the  piston  displacement.     Then,  if  expansion  in  the 

clearance  space  follows  the  law  pvn  =  PVn,  the  volume  of  clearance  air 
at  atmospheric  pressure  is 


of  the  piston  displacement.     For  the  displacement  above  given,  we  there- 
fore write, 


This   may  be  increased  5  to  10  per  cent,  to  allow  for  air  friction,  air 

heating,  etc. 

231.    Design  of  Compressor.     The  following  data  must  be  assumed  : 

(a)  capacity,  or  piston  displacement, 

(b)  maximum  pressure, 

(c)  initial  pressure  and  temperature, 

(d)  temperature  of  cooling  water, 

(e)  gas  to  be  compressed,  if  other  than  air. 

Let  the  compressor  deliver  300  cu.  ft.  of  compressed  air,  measured 
at  70°  F.,  per  minute,  against  100  Ib.  gauge  pressure,  drawing  its  supply  at 
14.7  Ib.  and  70°  F.,  the  clearance  being  2  per  cent.  Then,  ideally,  the  free 


114 


APPLIED  THERMODYNAMICS 


air  per  minute  will  be  300  x  (114.7  -^  14.7)  =  2341  cu.  ft,  or  allowing  5 
per  cent  for  losses  due  to  air  friction  and  heating  during  the  suction, 
2341  -*-  0.95  =  2464  cu.  ft.  To  allow  for  clearance,  we  may  use  the  ex- 
pression in  Art.  230,  making  the  displacement,  with  adiabatic  expansion 
of  the  clearance  air, 

2464-=-  [1-0.02  AMiTjrk  +  o.02]  =  2640  cu.  ft. 

Assuming  for  a  compressor  of  this  capacity  a  speed  of  80  r.  p.  m.,  the 
necessary  piston  displacement  for  a  double-acting  compressor  is  then 
2640  -r-  (2  x  80)  =  16.5  cu.  ft.  per  stroke,  or  for  a  stroke  of  3  ft.,  the  piston 
area  would  be  792  sq.  in.  (13).  The  power  expended  for  any  assumed 
compressive  path  may  be  calculated  as  in  Art.  190,  and  if  the  mechanical 
efficiency  be  assumed,  the  power  necessary  to  drive  the  compressor  at 
once  follows.  The  assumption  of  clearance  as  2  per  cent  must  be  justified 
in  the  details  of  the  design.  The  elevation  in  temperature  of  the  air  may 
be  calculated  as  in  Art.  185,  and  the  necessary  amount  of  cooling  water 
as  in  Art.  203,  the  exponents  of  the  curves  being  assumed. 

232.  Two-stage  Compressor.     From  Art.  211  we  may  establish  an  inter- 
mediate pressure  stage.     This  leads  to  a  new  correction  for  clearance,  and 
to  a  smaller  loss  of  capacity  due  to  air  heating.     Using  these  new  values, 
we  calculate  the  size  of  the  first-stage  cylinder.     For  the  second  stage,  the 
maximum  volume  may  be  calculated  on  the  basis  that  intercooling  is  com- 
plete, whence  the  cylinder  volumes  are  inversely  proportional  to  the  suc- 
tion pressures.     The  clearance  correction  will  be  found  to  be  the  same  as 
in  the  low-pressure  cylinder.     The  capacity,  temperature  rise,  water  con- 
sumption, power  consumption,  etc.,  are  computed  as  before.     A  considera- 
ble saving  in  power  follows  the  change  to  two  stages. 

233.  Problem.     Find  the  cylinder  dimensions  and  power  consumption  of  a 
double-acting  single-stage  air  compressor  to  deliver  4000  cu.  ft.  of  free  air  per 

minute  at  100  Ib.  gauge  pres- 
sure at  80  r.  p.  m.,  the  intake 
air  being  at  13.7  Ib.  absolute 
pressure,  the  piston  speed 
640  ft.  per  minute,  clearance 
4  per  cent,  and  the  clearance- 
expansion  and  compression 
curves  following  the  law 
PV^  =  c. 

Lay  off  the  distance  (777, 
Fig.  84,  to  represent  the  (un- 
known) displacement   of  the 
Art.  233.  — Design  of  Compressor.  piston,  which  we  will  call  D. 


COMPRESSOR  DESIGN 


115 


Since  the  clearance  is  4  per  cent,  lay  off  GZ  =  0.04  />,  determining  ZT  as  a 
coordinate  axis.  Draw  the  lines  TU,  VW,  YX,  representing  the  absolute  pres- 
sures indicated.  The  compression  curve  CE  may  now  be  drawn  through  C,  and 
the  clearance  expansion  curve  DI  through  D.  The  ideal  indicator  diagram  is 
CEDI.  We  have, 

/114.774 

"  D  ™   I     .. I 


or    Kr=i— * 


or  V= 


or  VA  = 


or  F*  = 


13.7  J 


0.04  D=  0.1927 


1.04  />=  0.2158  D. 


0.04  D  =  0.1829  i>. 


1.04  D=  0.9872  />. 


EutAB=  VB—  FA  =  0. 8043  D  is  the  volume  of  free  air  drawn  into  the  cylinder: 
A B-t-GH =0.8043  is  the  volumetric  efficiency :  to  compress  4000  cu.  ft.  of  free  air  per 
minute  the  piston  displacement  must  then  be  4:000  +  0.8043  =  4973  cu.  ft.  per  minute. 
Since  the  compressor  is  double-acting,  the  necessary  cylinder  area  is  the  quotient 
of  displacement  by  piston  speed  or  4973  -r-  640,  giving  7.77  sq.  ft.,  or  (neglecting 
the  loss  of  area  due  to  the  piston  rod),  the  cylinder  diameter  is  37.60  in.  From  the 
conditions  of  the  problem,  the  stroke  is  640  -4-  (2  x  80)  =  4ft. 

For  the  power  consumption,  we  have 

W=  GDEF+  FECH  -  JICH  -  GDIJ 

p    y     _  p    y  P    V     P  V 


=  PE(VE- 


0.35 


=  144^(114.7x0.1758)+ 


0.35 
x  0.2158) -(13.7  x  1.04) 


-(13.7x0.8473)- 


0.35 

(114.7  x  0.04) -(13.7  x  0.1927)-| 
0.35  J 

=  144  D  [20.16  +  30.01  -  11.61  -  5.59]  =  144  D  x  32.97. 

This  is  the  work  for  a  piston  displacement  =  D  cubic  feet.     If  we  take  D  at  4973 
per    minute,   the    horse   power 
consumed  in  compression  is 


144  x  32.97  x  4973  = 
33000 


1U.7 


3 


JF 


234.  Design  of  a  Two- 
stage  Machine.  With  condi- 
tions as  in  the  preceding,  con- 
sider a  two-stage  compressor 
with  complete  intercooling  and 
a  uniform  friction  of  one  pound 
between  the  stages.  Here  the 
combined  diagrams  appear  as 
in  Fig.  85.  For  economy  of 
power,  the  intermediate  pres-  FIG.  85.  Art.  231.—  Design  of  Two-stage  Compressor. 


J 


\ 


116  APPLIED  THERMODYNAMICS 


sure  is  V114.7  x  13.7  =  39.64,  whence  the  first-stage  discharge  pressure  and  the 
second-stage  suction  pressure,  corrected  for  friction,  are  respectively  40.14  and 
39.14  Ib.  For  the/rs*  stage,  Fig.  85, 

PF  =  PG  =  40.14,  PA  =  PB  =  14.7,  Pq  =  PB  =  13.7,  Va  =  1.04  A  VP  =  0.04  D. 
PGVG^  =  PHVH^  or  VG  =       !\M*VH  =   -'741.04  D  =  0.4701  D. 


or  Vq  =  V,  =  0.04  D  =  0.08864  D. 

f  ql  \  1O.7   / 

'74  '0.04/>=  0.08412  D. 


PAFA1-85  =  PjJV'86  or  VA  =  (^Y'74  VP=  j^y  V'740. 

\PA'  \  14.7  / 

PBVJM  =  PHVJ-»  or  V,  =  f~)°*TVa  -  (fff  )L04  D  =  0-987  D. 


The  volumetric  efficiency  is  AB  +  D  =  (VB  -  VA)  +D  =  0.987  -  0.08412  =  0  .-90288. 
Tine  piston  displacement  per  minute  is  4000  •+•  0.903  =  4430.  The  piston  diameter 
is  V(4430  -*-  640)  x  144  -  0.7854  =  35.6  in.  for  a  rfro&e  of  630  -r-  (2  x  80)  =  4ft. 
The  power  consumption  for  this  first  stage  is, 

W  =  PG(  VG  -  F,)  +  PG  VG  ~  P"Va  -  Pa(VH  -  Fg)  -  PpVF 


n  —  1  n  —  1 

=  [40.14(0.4701  -  0.04)  +  (40-U  x  0.4701)  -(13.7  x  1.04) 

0.35 

-  13.7(1.04  -  0.0886)  -  (40.14  X  0.04)  -(18.7  X  0.0886)-|144  D 

0.35  J 

=  2348.64  D  foot-pounds  or   10,404,475  foot-pounds  per  minute,  equivalent  to 
315.3  horse  power. 

SECOND  STAGE 

k 

Complete  interceding  means  that  at  the  beginning  of  compression  in  the  sec- 
ond stage  the  temperature  of  the  air  will  be  as  in  the  first  stage,  70°  F.     The 


P 

volume  at  this  point  will  then  be  Vz  =  =-£  Vn  =         -  IMD=  0.364  D.    We  thus 

±2  o9.14 

locate  the  point  Z,  Fig.  85,  and  complete  the  diagram  ZCEI,  making  VE  =  0.04 
(  Vz  -  VE)  =0.01456  />,  PC=PE  =  114.7,  P,  =  PZ  =  39.14,  and  compute  as  follows  : 


*  or  VK  =       f         Vz  =  M4D=  0.3574  D. 

PKI  V40.14/ 


or  Vc  -^  =  0.364Z)=  0.1642  D. 

P 


=  PEVE^  or  Vj=  Vm  =    |  0-0146  D  =  0.0318  D. 

P77,i.»  =  p^  TV35  or  F>  =  (  ?/]  °'74  VE  =  (  <—  )  °'74  0.0146  D  =  0.0324  D. 

\  -L  f  /  Noy.iT/ 


COMPRESSOR  DESIGN  117 

The  piston  displacement  is  Vz  —  VB  =  0.3494  D',  the  volumetric  efficiency  is  the  quo- 
tient of  (VK-  V  )  =  0.3256  D  by  this  displacement,  or  0.932.  For  a  stroke  of  4ft., 
the  cylinder  diameter  is  V[(0.3491  D  =  1547.84)  -  640]  x  144  --  0.7854  =  21.06  in. 
The  power  consumption  for  this  stage  is 

[(114.7  x  0.1496)  +  (H*-7x0.1642)  -(39.14x0.364) 

' 


W  = 

-(39.14  x  0.3316)-  <U"  x  00146)  ~  j88-1*  x  0-0324>"| 

0.35  J 

=  315.03  horse  power. 

The  total  horse  power  for  the  two-stage  compressor  is  then  630.33,  and  (within 
the  limit  of  the  error  of  computation)  the  work  is  equally  divided  between  the 

stages. 

235.  Comparisons.     We  note  then,  that  in  two-stage  compression,  the  saving 

in  power  is  —  —  ^—  ^  —  '•  —  =  0.118  of  the  power  expended  in  single-stage  compres- 
715 

sion  ;  that  the  low-pressure  cylinder  of  the  two-stage  machine  is  somewhat  smaller 
than  the  cylinder  of  the  single-stage  compressor;  and  that,  in  the  two-stage 
machine,  the  cylinder  areas  are  (approximately)  inversely  proportional  to  the  suction 
pressures.  The  amount  of  cooling  water  required  will  be  found  to  be  several  times 
that  necessary  in  the  single-stage  compressor. 

236.  Power  Plant  Applications.     On  account  of  the  ease  of  solution  of  air  in 
water,  the  boiler  feed  and  injection  waters  in  a  power  plant  always  carry  a  con- 
siderable quantity  of  air  with  them.     The  vacuum  pump  employed  in  connection 
with  a  condenser  is  intended  to  remove  this  air,  as  well  as  the  water.     It  is  esti- 
mated that  the  waters  ordinarily  contain  about  -fa  of  their  volume  of  air  at  atmos- 
pheric pressure.     The  pump  must  be  of  size  sufficient  to  handle  this  air  when 
expanded  to  the  pressure  in  the  condenser.     Its  cycle  is  precisely  that  of  any  air 
compressor,  the  suction  stroke  being  at  condenser  pressure  and  the  discharge 
stroke  at  atmospheric  pressure.    The  water  present  acts  to  reduce  the  value  of  the 
exponent  n,  thus  permitting  of  fair  economy. 

237.  Dry  Vacuum  Pumps.     In  some  modern  forms  of  high  vacuum  apparatus, 
the  air  and  'water  are  removed  from  the  condenser  by  separate  pumps.     The 
amount  of  air  to  be  handled  cannot  be  computed  from  the  pressure  and  tempera- 
ture directly,  because  of  the  water  vapor  with  which  it  is  saturated.     From  Dai- 
ton's  law,  and  by  noting  the  temperature  and  pressure  in  the  condenser,  the  pressure 
of  the  air,  separately  considered,  may  be  computed.     Then  the  volume  of  air,  cal- 
culated as  in  Art.  236,  must  be  reduced  to  the  condenser  temperature  and  pressure, 
and  the  pump  made  suitable  for  handling  this  volume  (11). 

COMMERCIAL  TYPES  OF  COMPRESSING  MACHINERY 

238.  Classification  of  Compressors.     Air  compressors  are  classified  according 
to  the  number  of  stages,  the  type  of  frame,  the  kind  of  valves,  the  method  of 
driving,  etc.     Steam-driven  compressors  are  usually  mounted  as  one  unit  with  the 
steam  cylinders  and  a  single  common  fly  wheel.     Regulation  is  usually  effected  by 


118 


APPLIED  THERMODYNAMICS 


TYPES  OF   COMPRESSOR 


119 


varying  the  speed.  The  ordinary  centrifugal  governor  on  the  steam  cylinder  im- 
poses a  maximum  speed  limit;  the  shaft  governor  is  controlled  by  the  air  pressure, 
which  automatically  changes  the  point  of  cut-off  on  the  steam  cylinder.  Power- 
driven  compressors  may  be  operated  by  electric  motor,  belt,  water  wheel,  or  in 
other  ways.  They  are  usually  regulated  by  means  of  an  "  unloading  valve,"  which 
either  keeps  the  suction  valve  closed  during  one  or  more  strokes  or  allows  the  air 
to  discharge  into  the  atmosphere  whenever  the  pipe  lines  are  fully  supplied.  In 
air  lift  practice,  a  constant  speed  is  sometimes  desired,  irrespective  of  the  load. 
In  the  "  variable  volume  "  type  of  machine,  the  delivery  of  the  compressor  is 
varied  by  closing  the  suction  valve  before  the  completion  of  the  suction  stroke. 
The  air  in  the  cylinder  then  expands  below  atmospheric  pressure. 


239.  Standard  Forms.  The  ordinary  small  compressor  is  a  single-stage 
machine,  with  poppet  air  valves  on  the  sides  of  the  cylinder.  The  frame  is  of  the 
"  fork  "  pattern,  with  bored  guides,  or  of  the  "  duplex  "  type,  with  two  single-stage 
cylinders.  These  machines  maybe  either  steam  or  belt  driven.  The  "straight 
line  "  compressors  differ  from  the  duplex  in  having  all  of  the  cylinders  in  one 
straight  line,  regardless  of  their  number. 

For  high-grade  service,  in  large  units,  the  standard  form  is  the  cross-compound 
two-stage  machine,  the  low-pressure  steam  and  air  cylinders  being  located  tandem 
beside  the  high-pressure  cylinders,  and  the  air  cylinders  being  outboard,  as  in 
Fig.  86.  Ordinary  standard  machines  of  this  class  are  built  in  capacities  ranging 
up  to  6000  cu.  ft.  of  free  air  per  minute.  The  other  machines  are  usually  con- 
structed only  in  smaller  sizes,  ranging  down  to  as  small  as  100  cu.  ft.  per  minute. 

Some  progress  has  been  made  in  the  development  of  rotary  compressors  for 
direct  driving  by 
steam  turbines.  The 
efficiency  is  fully  as 
high  as  that  of  an 
ordinary  reciprocat- 
ing compressor,  and 
the  mechanical  losses 
are  much  less. 


240.    Hydraulic 
Piston   Compressors; 

Sommeiller's.    In  Fig.    '4^-E^^iEJ^^[ l^^^^£^=^~-^^^^i^^i^r 
87,  as   the  piston  F 
moves   to    the   right, 
air  is  drawn  through 
C    to    G,    to-       .„ 
gether     with 
cooling   water 

from  B.      On     ' 

t  ^  ,  FIG.  87.    Art.  240.  —  Sommeiller  Hydraulic  Piston  Compressor. 

stroke,  the  air  is  compressed  and  discharged  through  D  and  A.     Indicator  dia- 
grams are  given  in  Fig.  88. 


120 


APPLIED  THERMODYNAMICS 


The  value  of  n  is  exceptionally  low,  and  clearance  expansion  almost  elimi- 
nated.    This  was  the  first  commercial  piston  compressor,  and  it  is  still  used  to  a 


FIG.  88.    Art.  240.  —  Variable  Discharge  Pressure  Indicator  Diagrams,  Sommeiller 

Compressor. 

limited  extent  in  Europe,  the  large  volume  of  water  present  giving  effective  cool- 
ing. It  cannot  be  operated  at  high  speeds,  on  account  of  the  inertia  of  the 
water. 

The  Leavitt  hydraulic  piston  compressor  at  the  Calumet  and  Hecla  copper 
mines,  Michigan,  has  double-acting  cylinders  60  by  42  in.,  and  runs  at  25  revolu- 
tions per  minute,  a  comparatively 
high  speed.  The  value  of  n  from  the 
card  shown  in  Fig.  89  is  1.23. 

241.    Taylor  Hydraulic  Compressor. 

Water  is  conducted  through  a  vertical 
shaft  at  the  necessary  head  (2.3  ft.  per 
pound  pressure)  to  a  separating  cham- 


FIG.  89.     Art.   248.  — Cards  from  Leavitt 
Compressor. 


FIG.  90.     Art.  241.  —  Taylor  Hydraulic 
Compresssor. 


TYPES  OF  COMPRESSOR 


121 


her.  The  shaft  is  lined  with  a  riveted  or  cast-iron  cylinder,  and  at  its  top  is  a 
dome,  located  so  that  the  water  flows  downward  around  the  inner  circumference 
of  the  cylinder.  The  dome  is  so  made  that  the  water  alternately  contracts  and 
expands  during  its  passage,  producing  a  partial  vacuum,  by  means  of  which  air  is 
drawn  in  through  numerous  small  pipes.  The  air  is  compressed  at  the  tempera- 
ture of  the  water  while  descending  the  shaft.  The  separating  chamber  is  so 
large  as  to  permit  of  separation  of  the  air  under  an  inverted  bell,  from  which  it  is 
led  by  a  pipe.  The  efficiency  is  0.60  to  0.70,  some  air  being  always  carried  away 
in  solution.  The  initial  cost  is  high,  and  the  system  can  be  installed  only  where 
a  head  of  water  is  available.  Figure  90  illustrates  the  device  (15). 

242.  Details  of  Construction.  The  standard  form  of  cylinder  for  large  machines 
is  a  two-piece  casting,  the  working  barrel  being  separate  from  the  jacket,  so  that 
the  former  may  be  a  good  wearing  metal  and  may  be  quite  readily  removable. 
Access  to  the  jacket  space  is  provided  through  bolt  holes. 

On  the  smaller  compressors,  the  poppet  type  of  valve  is  frequently  used  for  both 
inlet  and  discharge  (Fig.  91).  It  is  usually  considered  best  to  place  these  valves 


FIG.  91. 


Art.  242.— Compressor  Cylinder  with  Poppet  Valves. 
(Clayton  Air  Compressor  Works.) 


in  the  head,  thus  decreasing  the  clearance.  They  are  satisfactory  valves  for  auto- 
matically controlling  the  point  of  discharge,  excepting  that  they  are  occasionally 
noisy  and  uncertain  in  closing.  Poppet  valves  work  poorly  at  very  low  pressures, 
and  are  not  generally  used  for  controlling  the  intake  of  air.  Some  form  of 
mechanically  operated  valve  is  preferably  employed,  such  as  the  semi-rocking  type 


122 


APPLIED  THERMODYNAMICS 


of  Fig.  92,  located  at  the  bottom  of  the  cylinder,  which  has  poppet  valves  for  the 
discharge  dt  the  top.  For  large  units,  Corliss  inlet  valves  are  usually  employed, 

these  being  rocking  cyl- 
indrical valves  running 
crosswise.  As  in  steam 
engines,  they  are  so 
driven  from  an  eccentric 
and  wrist  plate  as  to  give 
rapid  opening  and  closing 
of  the  port,  with  a  com- 
paratively slow  interven- 
ing movement.  They  are 
not  suitable  for  use  as 
discharge  valves  in  single- 
stage  compressors,  or  in 
the  high-pressure  cylin- 
ders of  multi-stage  com- 
pressors, as  they  become 
fully  open  too  late  in  the 
stroke  to  give  a  suffi- 
ciently free  discharge. 
In  Fig.  93  Corliss  valves 
are  used  for  both  inlet 
and  discharge.  The 
auxiliary  poppet  shown 
is  used  as  a  safety  valve. 

A  gear  sometimes  used  consists  of  Corliss  inlet  valves  and  mechanically  operated 
discharge  valves,  which,  latter,  though  expensive,  are  free  from  the  disadvantages 
sometimes  experienced  with  poppet  valves.  The  closing  only  of  these  valves  is 
mechanically  controlled.  Their  opening  is  automatic. 


FIG.  92. 


SUCTJON 

Art.  242.  —  Compressor  Cylinder  with  Rocking  Inlet 
Valves.     (Clayton  Air  Compressor  Works.) 


FIG.  93.    Art.  242.  —  Compressor  Cylinder  with  Corliss  Valves.    (Allis-Chalmers  Co.) 


COMPRESSED  AIR  TRANSMISSION  123 


COMPRESSED  AIR  TRANSMISSION 

243.  Transmissive  Losses.     The  air  falls  in  temperature  and  pressure  in  the 
pipe  line.     The  fall  in  temperature  leads  to  a  decrease  in  volume,  which  is  further 
reduced  by  the  condensation  of  water  vapor ;  the  fall  in  pressure  tends  to  increase 
the  volume.     Early  experiments   at  Mont   Cenis  led  to  the  empirical  formula 
F  =  0.00000936  (n2/  -=-  d),  for  a  loss  of  pressure  F  in  a  pipe  d  inches  in  diameter, 
I  ft.  long,  in  which  the  velocity  is  n  feet  per  second  (16). 

In  the  Paris  distributing  system,  the  main  pipe  was  300  mm.  in  diameter,  and 
about  |  in.  thick,  of  plain  end  cast  iron  lengths  connected  with  rubber  gaskets. 
It  was  laid  partly  under  streets  and  sidewalks,  and  partly  in  sewers,  involving  the 
use  of  many  bends.  There  were  numerous  drainage  boxes,  valves,  etc.,  causing 
resistance  to  the  flow ;  yet  the  loss  of  pressure  ranged  only  from  3.7  to  5.1  lb.,  the 
average  loss  at  3  miles  distance  being  about  4.4  lb.,  these  figures  of  course  including 
leakage.  The  percentage  of  air  lost  by  leakage  was  ascertained  to  vary  from  0.38 
to  1.05,  including  air  consumed  by  some  small  motors  which  were  unintentionally 
kept  running  while  the  measurements  were  made.  This  loss  would  of  course  be 
proportionately  much  greater  when  the  load  was  light. 

244.  Unwin's  Formula.     Unwin's   formula   for   terminal  pressure  after  long 
transmission  is  generally  employed  in  calculations  for  pipe  lines  (17).     It  is, 


p  = 

iii  which  p  —  terminal  pressure  in  pounds  per  square  inch, 
P  =  initial  pressure  in  pounds  per  square  inch, 
/  =  an  experimental  coefficient, 
u  =  velocity  of  air  in  feet  per  second, 
L  =  length  of  pipe  in  feet, 
d  =  diameter  of  (circular)  pipe  in  feet, 
T  =  absolute  temperature  of  the  air,  F°. 

A  simple  method  of  determining/is  to  measure  the  fall  of  pressure  under  known 
conditions  of  P,  u,  T,  Z,  and  d,  and  apply  the  above  formula.  Unwin  has  in  this 
way  rationalized  the  results  of  Riedler's  experiments  on  the  Paris  distributing 
system,  obtaining  values  ranging  from  0.00181  to  0.00449,  with  a  mean  value 
f—  0.00290.  For  pipes  over  one  foot  in  diameter,  he  recommends  the  value  C.003 ; 
for  6-inch  pipe,/=  0.00435;  for  8-inch  pipe,/=  0.004. 

Riedler  and  Gutermuth  found  it  possible  to  obtain  pipe  lengths  as  great  as 
10  miles  in  their  experiments  at  Paris.  Previous  experiments  had  been  made,  on 
a  smaller  scale,  by  Stockalper.  For  cast-iron  pipe,  a  harmonization  of  these 
experiments  gives /=  0.0027(1  +  0.3  d),  d  being  the  diameter  of  the  pipe  in  feet. 
The  values  of  f  for  ordinary  wrought  pipe  are  probably  not  widely  different.  In 
any  well-designed  plant,  the  pressure  loss  may  be  kept  very  low. 

245.  Storage  of  Compressed  Air.  Air  is  sometimes  stored  at  very  high  pres- 
sures for  the  operation  of  locomotives,  street  cars,  buoys,  etc.  An  important  con- 


124  APPLIED  THERMODYNAMICS 

sequence  of  the  principle  illustrated  in  Joule's  porous  plug  experiment  (Art.  74) 
here  comes  into  play.  It  was  remarked  in  Art.  74  that  a  slight  fall  of  temperature 
occurred  during  the  reduction  of  pressure.  This  was  expressed  by  Joule  by  the 
formula 


in  which  F  was  the  fall  of  temperature  in  degrees  Centigrade  for  a  pressure 
drop  of  100  inches  of  mercury  when  T  was  the  initial  absolute  temperature 
(Centigrade)  of  the  air.  For  air  at  70°  F.,  this  fall  is  only  1^°  F.,  but  when  stored 
air  at  high  pressure  is  expanded  through  a  reducing  valve  for  use  in  a  motor,  the 
pressure  change  is  frequently  so  great  that  a  considerable  reduction  of  tempera- 
ture occurs.  The  efficiency  of  the  process  is  very  low ;  Peabody  cites  an  instance 
(18)  in  which  with  a  reservoir  of  75  cu.  ft.  capacity,  carrying  450  Ib.  pressure, 
with  motors  operating  at  50  Ib.  pressure  and  compression  in  three  stages,  the 
maximum  computed  plant  efficiency  is  only  0.29.  An  element  of  danger  arises  in 
compressed  air  storage  plants  from  the  possibility  of  explosion  of  minute  traces 
of  oil  at  the  high  temperatures  produced  by  compression. 

246.  Liquefaction  of  Air  ;  Linde  Process  (19).     The  fall  of  temperature  accom- 
panying a  reduction  of  pressure  has  been  utilized  by  Linde  and  others  in  the 
manufacture  of  liquid  air.      Air  is  compressed  to  about  2000  Ib.  pressure  in  a 
three-stage  machine,  and  then  delivered  to  a  cooler.     This  consists  of  a  double 
tube  about  400  ft.  long,  arranged  in  a  coil.     The  air  from  the  compressor  passes 
through  the  inner  tube  to  a  small  orifice  at  its  farther  end,  where  it  expands  into 
a  reservoir,  the  temperature  falling,  and  returns  through  the  outer  tube  of  the 
cooler  back  to  the  compressor.     At  each  passage,  a  fall  of  temperature  of  about 
37^°  C.  occurs.     The  effect  is  cumulative,  and  the  air  soon  reaches  a  temperature 
at  which  the  pressure  will  cause  it  to  liquefy  (Art.  610). 

247.  Refrigeration  by  Compressed  Air.     This  subject  will  be  more  particularly 
considered  in  a  later  chapter.     The  fall  of  temperature  accompanying  expansion 
in  the  motor  cylinder,  with  the  difficulties  which  it  occasions,  have  been  men- 
tioned in  Art.  185.     Early  in  the  Paris  development,  this  drop  of  temperature  was 
utilized  for  refrigeration.     The  exhaust  air  was  carried  through  flues  to  wine 
cellars,  where  it  served  for  the  cooling  of  their  contents,  the  production  of  ice,  etc. 
In  some  cases,  the  refrigerative  effect  alone  is  sought,  the  performance  of  work 
during  the  expansion  being  incidental. 

(1)  Riedler,  Neue  Erfahrungen  iiber  die  Kraftversorgung  von  Paris  durc.li  Druck- 
hift:  Berlin,  1891.  (2)  Pernolet  (L' Air  Comprime)  is  the  standard  reference  on  this 
work.  (3)  Experiments  upon  Transmission,  etc.  (Idelled.),  1903,  98.  (4)  Unwin,  op. 
cit.,  18  et  seq.  (5)  Unwin,  op.  cit.,  32.  (6)  Graduating  Thesis,  Stevens  Institute  of 
Technology,  1891.  (7)  Unwin,  op.  cit.,  48.  (8)  Op.  cit.,  109.  (9)  Unwin,  op.  cit.,  48,  49; 
some  of  the  final  figures  are  deduced  from  Kennedy's  data.  (10)  Power,  February  23, 
1909,  p.  382.  (11)  Development  and  Transmission  of  Power,  182.  (12)  Engineering 
News,  March  19,  1908,325.  (13)  Peabody,  Thermodynamics,  1907,  378.  (14)  Ibid., 
375.  (15)  Hiscox,  Compressed  Air,  1903,  273.  (16)  Wood,  Thermodynamics,  1905, 
306.  (17)  Transmission  by  Compressed  Air,  etc.,  68 ;  modified  as  by  Peabody. 


COMPRESSED   AIR  125 

(18)    Thermodynamics,    1907,   393,   394.      (19)  Zeuner,    Technical    Thermodynamics 
(Klein)  ;  II,  303-313 :    Trans.  A.  S.  M.  E.,  XXI,  156. 


SYNOPSIS   OF  CHAPTER  IX 

The  use  of  compressed  cold  air  for  power  engines  and  pneumatic  tools  dates  from  1860. 

The  Air  Engine 

The  ideal  air  engine  cycle  is  bounded  by  two  constant  pressure  lines,  one  constant 
volume  line,  and  a  poly  tropic.  In  practice,  a  constant  volume  drop  also  occurs 
after  expansion. 

Work  formulas  : 

pv+pv  loge  Z_  qv>,  pv+pv~PV_  qv-  pv  log,-;  (pv-PV}  (-X-  \ 
n  —  \  v  \y-lj 

Preheaters  prevent  excessive  drop  of  temperature  during  expansion  ;  the  heat  em- 

ployed is  not  wasted. 

Cylinder  volume  =  33,000  NEt  H-  2  n  Up,  ignoring  clearance. 
To  ensure  quiet  running,  the  exhaust  valve  is  closed  early,  the  clearance  air  acting  as  a 

cushion.    This  modifies  the  cycle. 

Early  closing  of  the  exhaust  valve  also  reduces  the  air  consumption. 
Actual  figures  for  free  air  consumption  range  from  400  to  2400  CM.  ft.  per  Ihp-hr. 

The  Compressor 

The  cycle  differs  from  that  of  the  engine  in  having  a  sharp  "  toe"  and  a  complete  clear- 

ance expansion  curve. 
Economy  depends  largely  on  the  shape  of  the  compression  curve.    Close  approximation 

to  the  isothermal,  rather  than  the  adiabatic,  should  be  attained,  as  during  expan- 

sion in  the  engine.     This  is  attempted  by  air  cooling,  jet  and  spray  injection  of 

water,  and  jacketing.     Water  required^  C—  H-^-  (S—  s)  . 

H(heatto  l,eab*tracted)  =  ^\(P}^-l}  +  -^-PV-( 
n  —  lL\P/  J     y  —  I     y—l\p 

Multi-stage  operation  improves  the  compression  curve  most  notably  and  is  in  other 

respects  beneficial. 
Intercooling  leads  to  friction  losses  but  is  essential  to  economy  ;  must  be  thorough. 


Work,  neglecting  clearance  (single  cylinder),  =  W=  ——A  (  —  |~  —  1    • 

n  —  1  L  \  PI  J 

The  area  under  the  compression  curve  is  called  the  work  of  compression. 
Minimum  work,  in  two-stage  compression,  is  obtained  when  P2  =  qp. 


Engine  and  Compressor  Relations 

Compressive  efficiency :  ratio  of  engine  work  to  compressor  work  ;  0.5  to  0.9. 
Mechanical  efficiency  :  ratio  of  work  in  cylinder  and  work  at  shaft ;  0.80  to  0.90. 
Cylinder  efficiency :  ratio  of  ideal  diagram  area  and  actual  diagram  area  ;  0.70  to  0.90. 
Plant  efficiency :  ratio  of  work  delivered  by  air  engine  to  work  expended  at  compressor 
shaft ;  0.25  to  0.45  ;  theoretical  maximum,  1.00. 


126  APPLIED  THERMODYNAMICS 

The  combined  ideal  entropy  diagram  is  bounded  by  two  constant  pressure  curves  and 
two  polytropics.  The  economy  of  thorough  intercooling  with  multi-stage  operation 
is  shown  ;  as  is  the  importance  of  a  low  exponent  for  the  polytropics.  With  very 
cold  water,  the  net  power  consumption  might  be  negative. 

Compressor  Capacity 

Volumetric  efficiency— ratio  of  free  air  drawn  in  to  piston  displacement ;  it  is  decreased 
by  excessive  clearance,  suction  friction,  heating  during  suction,  and  installation  at 
high  altitudes.  Long  stroke  compressors  have  proportionately  less  clearance. 
Water  may  be  used  to  fill  the  clearance  space :  multi-stage  operation  makes 
clearance  less  detrimental ;  refrigeration  of  the  entering  air  increases  the  volumet- 
ric efficiency.  Its  value  ranges  ordinarily  from  0.70  to  0.92.  Suction  friction 
and  clearance  also  decrease  the  cylinder  efficiency,  as  does  discharge  friction. 

Compressor  Design 

Theoretical  piston  displacement  per  stroke  =  -^ — ,  or  including  clearance, 

2  ntP 

to  be  increased  5  to  10  per  cent  in  practice. 
In  a  multi-stage  compressor  with  perfect  intercooling,  the  cylinder  volumes  are  inversely 

as  the  suction  pressures. 
The  power  consumed  in  compression  may  be  calculated  for  any  assumed  compressive 

path. 

A  typical  problem  shows  a  saving  of  12  per  cent  by  two-stage  compression. 
The  "  vacuum  pump"  used  with  a  condenser  is  an  air  compressor. 

Commercial  Types  of  Compressing  Machinery 

Classification  is  by  number  of  stages,  type  of  frame  or  valves,  or  method  of  driving. 
Governing  is  accomplished  by  changing  the  speed,  the  suction,  or  the  discharge  pressure. 
Commercial  types  include  the  single,  duplex,  straight  line  and  cross-compound  two-stage 

forms,  the  last  having  capacities  up  to  6000  cu.  ft.  per  minute.     Some  progress  has 

been  made  with  turbo-compressors. 

Hydraulic  piston  compressors  give  high  efficiency  at  low  speeds. 
The  Taylor  hydraulic  compressor  gives  efficiencies  up  to  0.60  or  0.70. 
Cylinder  barrels  and  jackets  are  separate  castings.     Access  to  water  space  must  be 

provided. 
Poppet,  mechanical  inlet,  Corliss,  and  mechanical  discharge  valves  are  used. 

Compressed  Air  Transmission 

Loss  in  pressure  =  0.00000936  nH  •*•  d. 

In  Paris,  the  total  loss  in  3  miles,  including  leakage,  was  4-4  lb. ;  the  percentage  of  leak- 
age was "0.38  to  1.05,  including  air  unintentionally  supplied  to  consumers. 

Unwin-s  formula;  jj-pfl  -  *u*  L  l|«   Mean  value  of  /=  0.0029.  /=  0.0027  (1+0.3(2). 
Fall  of  temperature  for  49  lb.  fall  of  pressure  by  throttling  =  0.92  f  SKV. 


COMPRESSED  AIR  127 

Stored  high  pressure  air  may  be  used  for  driving  motors,  but  the  efficiency  is  low. 
The  fall  of  temperature  induced  by  throttling  may  be  used  cumulatively  to  liquefy  air. 
The  fall  of  temperature  accompanying  expansion  in  the  engine  may  be  employed  for 
refrigeration. 

PROBLEMS 

1.  An  air  engine  works  between  pressures  of  180  Ib.  and  15  Ib.  per  square  inch, 
absolute.     Find  the  work  done  per  cycle  with  adiabatic  expansion  from  v  =  I  to  V  =  4, 
ignoring  clearance.     By  what  percentage  would  the  work  be  increased  if  the  expansion 
curve  were  PF1-3  =  c  ? 

2.  The  expansion  curve  is  PF13=c,  the  pressure  ratio  during  expansion  7  : 1,  the 
initial  temperature  100°  F.     Find  the  temperature  after  expansion.     To  what  tempera- 
ture must  the  entering  air  be  heated  if  the  final  temperature  is  to  be  kept  above  32°  F  ? 

3.  Find  the  cylinder  dimensions  for  a  double-acting  100  hp.  air  engine  with  clear- 
ance 4  per  cent,  the  exhaust  pressure  being  15  Ib.  absolute,  the  engine  making  200 
r.  p.  m.,  the  expansion  and  compression  curves  being  PF1-35  '=  c,  and  the  air  being 
received  at  160  Ib.  absolute  pressure.     Compression  is  carried  to  the  maximum  pres- 
sure, and  the  piston  speed  is  400  ft.  per  minute.     A  10-lb.  drop  of  pressure  occurs  at 
the  end  of  expansion.     (Allow  a  10  per  cent  margin  over  the  theoretical  piston  dis- 
placement.) 

4.  Estimate  the  free  air  consumption  per  Ihp.-hr.  in  the  engine  of  Problem  3. 

5.  A  hydrogen  compressor  receives  its  supply  at  70°  F.  and  atmospheric  pressure, 
and  discharges  it  at  100  Ib.  guage  pressure.     Find  the  temperature  of  discharge,  if  the 
compression  curve  is  PF1-32  =  c. 

6.  In  Problem  5,  what  is  the  percentage  of  power  wasted  as  compared  with  iso- 
thermal compression,  the  cycles  being  like  CBAD,  Fig.  57  ?     Consider  only  the  power 
necessary  to  compress  isothermally  to  the  maximum  pressure,  not  the  whole  power 
expended  in  the  cycle. 

7.  In  Problem  3,  find  what  quantity  of  heat  must  have  been  added  during  expan- 
sion to  make  the  path  PF1-35  =  c  rather  than  an  adiabatic.     Assuming  this  to  be  added 
by  a  water  jacket,  the  water  cooling  through  a  range  of  70°,  find  the  weight  of  water 
circulated  per  minute. 

8.  Find  the  receiver  pressures  for  minimum  work  in  two  and  four-stage  compres- 
sion of  atmospheric  air  to  guage  pressures  of  100,  125,  150,  and  200  Ib. 

•  9.  What  is  the  minimum  work  expenditure  in  the  cycle  compressing  free  air  at 
70°  F.  to  100  Ib.  guage  pressure,  per  pound  of  air,  along  a  path  PF1-35  =  c,  clearance 
being  ignored  ? 

10.  Find  the  cylinder  efficiency  in  Problem  3,  the  pressure  in  the  pipe  line  being 
165  Ib.  absolute. 

11.  Sketch  the  entropy  diagram  for  a  four-stage   compressor  and  two-stage  air 
engine,  in  which  n  is  1.3  for  the  compressor  and  1.4  for  the  engine,  the  air  is  inade- 
quately interceded,  perfectly  aftercooled,  and  inadequately  preheated  between  the 
engine  cylinders.     Compare  with  the  entropy  diagram  for  adiabatic  paths  and  perfect 
intercooling  and  such  preheating  as  to  keep  the  temperature  of  the  exhaust  above  32°  F. 


128  APPLIED  THERMODYNAMICS 

12.  Find  the  cylinder  dimensions  and  power  consumption  of  a  single-acting  single- 
stage  air  compressor  to  deliver  8000  cu.  ft.  of  free  air  per  minute  at  180  Ib.  absolute 
pressure  at  60  r.  p.  m.,  the  intake  air  being  at  13.0  Ib.  absolute  pressure,  the  piston  speed 
640  ft.  per  minute,  clearance  3  per  cent,  and  the  expansion  and  compression  curves  fol- 
lowing the  law  P  F"1-31  =  c. 

13.  With  conditions  as  in  Problem  12,  find  the  cylinder  dimensions  and  power 
consumption  if  compression  is  in  two  stages,  intercooling  is  perfect, and  2  Ib.  of  friction 
loss  occur  between  the  stages. 

14.  The  cooling  water  rising  from  68°  F.  to  89°  F.  in  temperature,  in  Art.  233, 
find  the  water  consumption  in  gallons  per  minute. 

15.  Find  the  water  consumption  for  jackets  and  intercooling  in  Art.  234,  the  range 
of  temperature  of  the  water  being  from  47°  to  68°  F. 

16.  Find  the  cylinder  volume  of  a  pump  to  maintain  26"  vacuum  when  pumping 
100  Ib.  of  air  per  minute,  the  initial  temperature  of  the  air  being  110°  F.,  compression 
and  expansion  curves  PF1-28  =  c,  clearance  6  per  cent.,  and  the  pump  having  two 
double-acting  cylinders.    The  speed  is  60  r.  p.  m.     Pipe  friction  may  be  ignored. 

17.  Compare  the  Riedler  and  Gutermuth  formula  for  /  (Art.  244)  with  Unwin's 
values. 

18.  In  a  compressed  air  locomotive,  the  air  is  stored  at  2000  Ib.  pressure  and  de- 
livered to  the  motor  at  100  Ib.     Find  the  temperature  of  the  air  delivered  to  the  motor 
if  that  of  the  air  in  the  reservoir  is  80°  F.,  assuming  that  the  value  of  F  (Art.  245)  is 
directly  proportional  to  the  pressure  drop. 

19.  With  isothermal  curves  and  no  friction,  transmission  loss,  or  clearance,  what 
would  be  the  combined  efficiency  from  compressor  to  motor  of  an  air  storage  system  in 
which  the  storage  pressure  was  450  Ib.  and  the  motor  pressure  50  Ib.?     The  tempera- 
ture of  the  air  is  80°  F.  at  the  motor  reducing  valve. 

20.  Find,  by  the  Mont  Cenis  formula,  the  loss  of  pressure  in  a  12-in.  pipe  2  miles 
long  in  which  the  air  velocity  is  32  ft.  per  second.     Compare  with  Unwin's  formula, 
using  the  Riedler  and  Gutermuth  value  for/,  assuming  P  =  80,  T  =  70°  F. 


CHAPTER   X 

HOT-AIR  ENGINES 

248.  General  Considerations.     From  a  technical  standpoint,  the  class  of 
air  engines  includes  all  heat  motors  using  any  permanent  gas  as  a  working 
substance.     For  convenience,  those  engines  in  which  the  fuel  is  ignited 
inside  the  cylinder  are  separately  discussed,  as  internal  combustion  or  gas 
engines  (Chapter  XI).     The  air  engine  proper  is  an  external  combustion 
engine,  although  in  some  types  the  products  of  combustion  do  actually 
enter  the  cylinder ;  a  point  of  disadvantage,  since  the  corrosive  and  gritty 
gases  produce  rapid  wear  and  leakage.     The  air  engine  employs,  usually, 
a  constant  mass  of  ivorking  substance)  i.e.  the  same  body  of  air  is  alter- 
nately heated  and  cooled,  none  being  discharged  from  the  cylinder  and  no 
fresh  supply  being  brought  in  ;  though  this  is  not  always  the  case.     Such 
an  engine  is  called  a  "  closed  "  engine.     Any  fuel  may  be  employed ;  the 
engines  require  little  attention ;  there  is  no  danger  of  explosion. 

Modern  improvements  on  the  original  Stirling  and  Ericsson  forms  of 
air  engine,  while  reducing  the  objections  to  those  types,  and  giving  excel- 
lent results  in  fuel  economy,  are,  nevertheless,  limited  in  their  application 
to  small  capacities,  as  for  domestic  pumping  service. 

In  air,  or  any  perfect  gas,  the  temperature  may  be  varied  independ- 
ently of  the  pressure ;  consequently,  the  limitation  referred  to  in  Art.  143 
as  applicable  to  steam  engines  does  not  necessarily  apply  to  aii\  engines, 
which  may  work  at  much  higher  initial  temperatures  than  any  steam  en- 
gine, their  potential  efficiency  being  consequently  much  greater.  When 
a  specific  cycle  is  prescribed,  however,  as  we  shall  immediately  find,  pres- 
sure limits  may  become  of  importance. 

249.  Capacity.     One  objection  to  the  air  engine  arises  from  the  ex- 
tremely slow  transmission  of  heat  through  metal  surfaces  to  dry  gases. 
This  is  partially  overcome  in  various  ways,  but  the  still  serious  objection 
is  the  small  capacity  for  a  given  size.     If  the  Carnot  cycle  be  plotted  for 
one  pound  of  air,  as  in  Fig.  94,  the  enclosed  work  area  is  seen  to  be  very 
small,  even  for  a  considerable  range  of  pressures.     The  isothermals  and 
adiabatics  very  nearly  coincide.     For  a  given  output,  therefore,  the  air  en- 
gine must  be  excessively  large  at  anything  like  reasonable  maximum  pres- 
sures.    In  the  Ericsson  engine  (Art.  269),  for  example,  although  the  cycle 

129 


130 


APPLIED   THERMODYNAMICS 


was  one  giving  a  larger  work  area  than  that  of  Carnot,  four  cylinders 
were  required,  each  having  a  diameter  of  14  ft.  and  a  stroke  of  6  ft. ;  it 
was  estimated  that  a  steam  engine  of  equal  power  would  have  required 


350 


P,00 


250 


200 


50 


\ 


=  1059.6  abs, 


1  2  3  4  5  6  7  8  9          10          11  12          13 

FIG.  94.    Arts.  249,  250.  —  Carnot  Cycle  for  Air. 

only  a  single  cylinder,  4  ft.  in  diameter  and  of  10-ft.  stroke,  running  at  17 
revolutions  per  minute  and  using  4  Ib.  of  coal  per  horse  power  per  hour. 
The  air  engine  ran  at  9  r.  p.  m.,  and  its  great  bulk  and  cost,  noisiness  and 
rapid  deterioration,  overbore  the  advantage  of  a  much  lower  fuel  consump- 
tion, 1.87  Ib.  of  coal  per  hp.-hr.  At  the  present  time,  with  increased 
steam  pressures  and  piston  speeds,  the  equivalent  steam  engine  would  be 
still  smaller. 


250.  Carnot  Cycle  Air  Engine.  The  efficiency  of  the  cycle  shown  in 
Fig.  94  has  already  been  computed  as  (T  —  £)-5-  T  (Art.  135).  The  work 
done  per  cycle  is,  from  Art.  135, 


POLYTROPIC   CYCLE  131 

Another  expression  for  the  work,  since 


But  from  Art.  104,  —2  = 


-1,  whence 


y-i 

andTF 
This  can  have  a  positive  value  only  when  —  if  —  V"1  exceeds  unity  ;  which 

PS\TJ 

p  /rp\J[_ 

is  possible  only  when  —  *  exceeds    —  *"1.     Now  since  Pl  and  P3  are  the 


limiting  pressures  in  the  cycle,  and  since  for  air  y  -s-  (y  —  1)  =  3.486,  the 
minimum  necessary  ratio  of  pressures  increases  as  the  3.486  power  of  the  ratio 
of  temperatures.*  This  alone  makes  the  cycle  impracticable.  In  Fig.  94, 
the  pressure  range  is  from  14.7  to  349.7  Ib.  per  square  inch,  although  the 
temperature  range  is  only  100*. 

251.  Polytropic  Cycle.  In  Pig.  95,  let  T,  t  be  two  isothermals,  eb  and  dftwo 
like  polytropic  curves,  following  the  law  pvH  =  c,  and  ed  and  bf  two  other  like 
polytropic  curves,  following  the  law  pvm  —  c. 
Then  ebfd  is  a  polytropic  cycle.  Let  T,  /,  Pb,  Pe 


be  given..    Then    Te  = 

T 


In  the  en- 
tropy diagram, 
Fig.  96,  locate  the 
isothermals  T,  t, 
Te.  Choose  the 
point  e  at  random. 
From  Art.  Ill,  the 
specific  heat  along 
a  path  pvn  =  c  is  FIG.  95.  Arts.  251,  256,  Prob.  4a. 
;  and 


n-l 

from  Art.    163,  the  increase  of   entropy  when  the 

specific    heat    is    s,    in   passing     from   e   to    b,   is 
FIG.  96.    Arts.  251,  256.  — Poly-  T 

tropic  Cycle.  N  =  s  loge  — .     This  permits  of  plotting  the  curve 

-I  e 


*  It  has  been  shown  that  —  =  (  A! 


But  P3  <  P4,  if  a  finite  work  area  is  to 


be  obtained  ;  hence     1  < 


132 


APPLIED  THERMODYNAMICS 


eb  in  successive  short  steps,  in  Fig.  96.     Along  ed,  similarly,  sl  =  // ^ )  and 

rr,  \m    —    */ 

JVj  =8^0%^  between  d  and  e.     We  complete  the  diagram  by  drawing  bf  and 

•*  d 

df,  establishing  the  point  of  intersection  which  determines  the  temperature  at  /. 


We  find  Tf:Tb::Td:Tt 

[nebx  +  xbfN  -  ydfN  -  nedy\ 
s(T  -  Te)+Sl 


ebfd 


The  efficiency  is  equal  to    r'?"   ,  or  to 


[nebx  +  xbfN] 


-  TV)  - 


s(Tb  -  Te)  +  8l(Tb-  T/) 


s(Tf- 


the  negative  sign  of  the  specific  heat  sl  being  disregarded. 

252.  Lorenz  Cycle.  In  Fig.  97  let  dg  and  bh  be  adiabatics,  and  let  the  curves 
gb  and  dh  be  polytropics,  but  unlike,  the  former  having  the  exponent  n,  and  the 
latter  the  exponent  q.  This  constitutes  the  cycle  of  Lorenz.  We  find  the  tempera- 


FIG.  97.    Arts.  252,  256,  Prob.  5.— 
Lorenz  Cycle. 


FIG.  98.    Arts.  252,  256. — Lorenz  Cycle, 
Entropy  Diagram. 


ture  at  g  as  in  Art.  251,  and  in  the  manner  just  described  plot  the  curves  gb  and 
dh  on  the  entropy  diagram,  Fig.  98,  Pff,  P5,  Tb,  Td,  n  and  q  being  given,  dg  and 
bh  of  course  appear  as  vertical  straight  lines.  The  efficiency  is 


n(Tb~  Tg) 


253.  Reitlinger  Cycle.  This  appears  as  aicj,  Figs.  99  and  100.  It  is  bounded 
by  two  isothermals  and  two  like  polytropics  (isodiabatics).  To  plot  the  entropy 
diagram,  Fig.  100,  we  assume  the  ratio  of  pressures  or  of  volumes  along  ai  or  cj. 

Let  Va  and  Vi  be  given.     Then  the  gain  of  entropy  from  a  to  i  is  f  Pa  Va  loge  —  J  -*-  T. 


JOULE  AIR  ENGINE 


133 


The  curves  ic  and  aj  are  plotted  for  the  given  value  of  the  exponent  n.     This  is 
sometimes  called  the  isod'uibatic  cycle.     Its  efficiency  is 

/  TT        ]      If  77"  IT     \  /  77"         I      IT    \ 

("m  +  flic  —  tljc  —  flaj)  +-   (_"ui  +  tlieji 

which  may  be  expanded  as  in  Arts.  251,  252. 

T 


FIG.  99.    Arts.  253, 256.— Reit- 
linger  Cycle. 


F,G.  100.  Arts.  253,  256,  257,  2> 
259.— Reitlinger  Cycle,  Entropy 
Diagram. 


254.  Joule  Engine.  An  air  engine  proposed  by  Ericsson  as  early  as 
1833,  and  revived  by  Joule  and  Kelvin  in  1851,  is  shown  in  Fig.  101.  A 
chamber  C  contains  air  kept  at  a  low  temperature  t  by  means  of  circulating 
water.  Another  chamber  A  contains  hot  air  in  a  state  of  compression, 
the  heat  being  supplied  at  a  constant  temperature  T  by  means  of  an  ex- 
ternal furnace  (not  shown).  M  is  a  pump  cylinder  by  means  of  which  air 


FIG.  101.     Arts.  25i,  255,  275. — Joule  Air  Engine. 

may  be  delivered  from  C  to  A,  and  N  is  an  engine  cylinder  in  which  air 
from  A  may  be  expanded  so  as  to  perform  work.  The  chambers  A  and  C 
are  so  large  in  proportion  to  M  and  N  that  the  pressure  of  the  air  in  these 
chambers  remains  practically  constant. 


134 


APPLIED  THERMODYNAMICS 


The  pump  M  takes  air  from  C,  compresses  it  adiabatically,  until  its 
pressure  equals  that  in  A,  then,  the  valve  v  being  opened,  delivers  it  to  A 

at  constant  pressure.  The  cycle 
is  /doe,  Fig.  102.  In  this  special 
modification  of  the  polytropic 
cycle  of  Art.  251,  fd  represents 
the  drawing  in  of  the  air  at  con- 
stant pressure,  do  its  adiabatic 
compression,  and  oe  its  discharge 
to  A.  Negative  work  is  done, 
equal  to  the  area  fdoe.  Concur- 
rently with  this  operation,  hot 


FIG.  102.    Arts.  254,  255,  256.— Joule  Cycle. 


air  has  been  flowing  from  A  to  N 
through  the  valve  u,  then  expand- 
ing adiabatically  while  u  is  closed ;  finally,  when  the  pressure  has  fallen 
to  that  in  (7,  being  discharged  to  the  latter  chamber,  the  cycle  being  ebqf, 
Fig.  102.  Positive  work  has  been  done,  and  the  net  positive  work  per- 
formed by  the  whole  apparatus  is  ebqf  —  fdoe  =  obqd. 


255.  Efficiency  of  Joule  Engine.  We  will  limit  our  attention  to  the  net 
cycle  obqd.  The  heat  absorbed  along  the  constant  pressure  line  ob  is 
H*  =  k(T  -  T0).  The  heat  rejected  along  qd  is  Hqd  =  Tc(Tq  -  t).  But 

T       T  T  _  t        t 

from  Art.  251,  —  ^  =  —  ,  whence,  -=±  —  w  =  -Ffl  ,  and  the  efficiency  is 

*  •*«  J-    —    *  *o 


Hoh-Hqd=         ^_1  _     Tq  -  t  _     _     t  _T0-t 
Ho,  Hob  T-T0~          T~     T0 

This  is  obviously  less  than  the 

efficiency  of  the  Carnot   cycle 

between  T  and  t.     The  entropy 

diagram  may  be  readily  drawn 

as   in   Fig.  103.      The   atmos- 

phere may  of  course  take  the 

place  of  the  cold  chamber    (7, 

a  fresh  supply  being  drawn  in 

by  the  pump  at  each  stroke,  and 

the    engine    cylinder    likewise 

discharging  its  contents  to  the 

atmosphere.    The  ratio  fd  -^  fq, 

in  Fig.  102,  shows  the  necessary  ratio  of  volumes  of  pump  cylinder  and 

engine  cylinder.     The  need  of  a  large  pump  cylinder  would  be  a  serious 

drawback  in  practice  ;  it  would  make  the  engine  bulky  and  expensive,  and 


FIG.  103. 


Arts.  255,  25G. — Joule  Cycle,  Entropy 
Diagram. 


REGENERATOR  135 

would  lead  to  an  excessive  amount  of  mechanical  friction.     The  Joule 
engine  has  never  been  constructed. 

256.  Comparisons.     The  cycles  just  described  have  been  grouped 
in  a  single  illustration  in  Fig.  104.     Here  we  have,  between  the 
temperature  limits  T  and  £,   the   Oarnot  cycle,  abed ;  the  polytropic 
cycle,   debf;    the    Lorenz 

cycle,  dgbh ;  that  of  Reit- 
linger,  aicj  \  and  that  of 
Joule,  obqd.  These  illus- 
trations are  lettered  to 
correspond  with  Figs. 
95-100,  102,  103.  A 
graphical  demonstration 
that  the  Carnot  cycle  is 
the  one  of  maximum 
efficiency  suggests  itself. 
We  now  consider  the 
most  successful  attempt 
yet  made  to  evolve  a  cycle 
having  a  potential  effi- 
ciency equal  to  that  of 
Carnot. 

257.  Regenerators. 
By  reference  to  Fig.  100, 
it  may  be  noted  that  the 
heat  areas  under  aj  and 
ic  are  equal.      The  heat 
absorbed  along   the    one 
path  is  precisely  equal  to 
that    rejected  along   the 
other.       This    fact    does 
not  prevent  the  efficiency 
from  being  less  than  that 
of  the  Carnot  cycle,   for 
efficiency  is  the  quotient 
of  work  done  by  the  gross 

heat   absorption.     If,    however,   the    heat   under  ic  were   absorbed 
not  from  the  working  substance,  and  that  under  ja  were  rejected 


FIG.  104.    Arts.  250,  266.  — Hot-air  Cycles. 


136  APPLIED  THERMODYNAMICS 

not  to  the  condenser  ;  but  if  some  intermediate  body  existed  having  a 
storage  capacity  for  heat,  such  that  the  heat  rejected  to  it  along  ja 
could  be  afterward  taken  up  from  it  along  ic,  then  we  might  ignore 
this  quantity  of  heat  as  affecting  the  expression  for  efficiency,  and  the 
cycle  would  be  as  efficient  as  that  of  Carnot.  The  intermediate  body 
suggested  is  called  a  regenerator. 


258.  Action  of  Regenerators.  Invented  by  Robert  Stirling  about  1816,  and 
improved  by  James  Stirling,  Ericsson,  and  Siemens,  the  present  form  of  regener- 
ator may  be  regarded  as  a  long  pipe,  the  walls  of  which  have  so  large  a  capacity 
for  heat  that  the  temperature  at  any  point  remains  practically  constant.  Through 
this  pipe  the  air  flows  in  one  direction  when  working  along  ic,  Fig.  100,  and 
in  the  other  direction  while  working  along  ja.  The  air  encounters  a  gradually 
changing  temperature  as  it  traverses  the  pipe. 

Let  hot  exhaust  air,  at  t,  Fig.  100,  be  delivered  at  one  end  of  the  regenerator. 
Its  temperature  begins  to  fall,  and  continues  falling,  so  that  when  it  leaves  the 
regenerator  its  temperature  is  that  at  c,  usually  the  temperature  of  the  atmosphere. 
The  temperature  at  the  inlet  end  of  the  regenerator  is  then  T,  that  at  its  outlet  t. 
During  the  admission  of  fresh  air,  along  ja,  it  passes  through  the  regenerator  in 
the  opposite  direction,  gradually  increasing  in  temperature  from  t  to  T,  without 
appreciably  affecting  the  temperature  of  the  regenerator.  Assuming  the  capacity  of 
the  regenerator  to  be  unlimited,  and  that  there  are  no  losses  by  conduction  of  heat 
to  the  atmosphere  or  along  the  material  of  the  regenerator  itself,  the  process  is 
strictly  reversible.  We  may  cause  either  the  volume  or  the  pressure  to  be  either 
fixed  or  variable  according  to  some  definite  law,  during  the  regenerative  move- 
ment. Usually,  either  the  pressure  or  the  volume  is  kept  constant. 

As  actually  constructed,  the  regenerator  consists  of  a  mass  of  thin  perforated 
metal  sheets,  so  arranged  as  not  to  obstruct  the  flow  of  air.  Some  waste  of  heat 
always  accompanies  the  regenerative  process ;  in  the  steamer  Ericsson,  it  was  10 
per  cent  of  the  total  heat  passing  through.  Siemens  appears  to  have  reduced  the 
loss  to  5  per  cent. 


259.  Influence  on  Efficiency.  Any  cycle  bounded  by  a  pair  of 
isothermals  and  a  pair  of  like  polytropics,  if  worked  with  a  regener- 
ator, has  an  efficiency  ideally  equal  to  that  of  the  Carnot  cycle.  To 
be  sure,  the  heated  air  is  not  all  taken  in  at  T,  nor  all  rejected  at  t\ 
but  the  heat  absorbed  from  the  source  is  all  at  T,  and  that  rejected 
to  the  condenser  is  all  at  t.  The  regenerative  operations  are  mutu- 
ally compensating  changes  which  do  not  affect  the  general  principle 
of  efficiency  under  such  conditions.  The  heat  paid  for  is  only  that 
under  the  line  ai,  Fig.  100.  The  regenerator  thus  makes  the  effi- 
ciency of  the  Carnot  cycle  obtainable  by  actual  heat  engines. 


THE  STIRLING  ENGINE 


137 


As  will  appear,  the  cycles  in  which  a  regenerator  is  commonly  employed  are 
not  otherwise  particularly  efficient.  Their  chief  advantage  is  in  the  large  work 
area  obtained,  which  means  increased  capacity  of  an  engine  of  given  dimensions. 
For  highest  efficiency,  the  regenerator  must  be  added. 


260.  The  Stirling  Engine.  This  important  type  of  regenerative  air  engine 
was  covered  by  patents  dated  1827  and  1840,  by  Robert  and  James  Stirling.  Its 
action  is  illustrated  in  Fig.  105.  G  is  the  engine 
cylinder,  containing  the  piston  H,  and  receiving 
heated  air  through  the  pipe  F  from  the  vessel  .4^4 
in  which  the  air  is  alternately  heated  and  cooled. 
The  vessel  A  A  is  made  with  hollow  walls,  the  inner 
lining  being  marked  aa.  T^he  hemispherical  lower 
portion  of  the  lining  is  perforated;  while  from  A  A 
up  to  CC  the  hollow  space  constitutes  the  regener- 
ator, being  filled  with  strips  of  metal  or  glass.  The 
plunger  E  fits  loosely  in  the  machined  inner  shell 
aa.  This  plunger  is  hollow  and  filled  with  some 
non-conducting  material.  The  spaces  DD  contain 
the  condenser,  consisting  of  a  coil  of  small  copper 
pipe,  through  which  water  is  circulated  by  a  sepa- 
rate pump.  An  air  pump  discharges  into  the  pipe 
F  the  necessary  quantity  of  fresh  air  to  compensate 
for  any  leakage,  and  this  is  utilized  in  some  cases 
to  maintain  a  pressure  which  is  at  all  stages  con- 
siderably above  that  of  the  atmosphere.  The  furnace  is  built  about  the  bottom 
ABA  of  the  heating  vessel. 


FIG.  105.      Arts.  260,  261,  262, 
263,  264.  — Stirling  Engine. 


261.  Action  of  the  Engine.  Let  the  plunger  E  and  the  piston  H  be  in  their 
lowest  positions,  the  air  above  E  being  cold.  The  plunger  E  is  raised,  causing 
air  to  flow  from  X  downward  through  the  regenerator  to  the  space  b,  while  H 
remains  motionless.  The  air  takes  up  heat  from  the  regenerator,  increasing  its 
temperature,  say  to  T,  while  the  volume  remains  constant.  After  the  plunger  has  come 
to  rest,  the  piston  H  is  caused  to  rise  by  the  expansion  produced  by  the  absorption 
of  heat  from  the  furnace  at  constant  temperature,  the  air  reaching  H  by  passing 
around  the  loose-fitting  plunger  E,  which  remains  stationary.  H  now  pauses  in 
its  "up"  position,  while  E  is  lowered,  forcing  air  through  the  regenerator  from 
the  lower  space  b  to  the  upper  space  A',  this  air  decreasing  in  temperature  at  con- 
stant volume.  While  E  remains  in  its  "down"  position,  H  descends,  forcing  the 
air  to  the  condenser  D,  the  volume  decreasing,  but  the  temperature  remaining  con- 
stant at  t.  The  cycle  is  thus  completed. 

The  working  air  has  undergone  four  changes  :  (a)  increase  of  pressure 
and  temperature  at  constant  volume,  (b)  expansion  at  constant  tempera- 
ture, (c)  a  fall  of  pressure  and  temperature  at  constant  volume,  and  (d) 
compression  at  constant  temperature. 


138 


APPLIED  THERMODYNAMICS 


262.  Remarks.     With  action  as  described,  the  piston  H  and  the  plunger  E 
(sometimes  called  the  "  displacer  piston  ")  do  not  move  at  the  same  time ;  one  is 
always  nearly  stationary,  at  or  near  the  end  of  its  stroke,  while  the  other  moves. 
In  practice,  uniform  'rotative  speed  is  secured  by  modifying  these  conditions,  so 
that  the  actual  cycle  merely  approximates   that   described.     The  vessel  A  A    is 
sometimes  referred  to  as  the  "receiver."     It  is  obvious  that  a  certain  residual 
quantity  of  air  is  at  all  times  contained  in  the  spaces  between  the  piston  H  and 
the  plunger  E.     This  does  not  pass  through  the  regenerator,  nor  is  it  at  any  time 
subjected  to  the  heat  of  the  furnace.     It  serves  merely  as  a  medium  for  transmit- 
ting pressure  from  the  "working  air"  to  H]    and  in  contradistinction  to  that 
working  substance,  it  is  called  "  cushion  air."     Being  at  all  times  in  communica- 
tion with  the  condenser,  its  temperature  is  constantly  close  to  the  minimum  attained  in 
the  cycle.     This  is  an  important  point  in  facilitating  lubrication. 

263.  Forms  of  the  Stirling  Engine.     In  some  types,  a  separate  pipe  is  carried 
from  the  lower  part  of  the  receiver  to  the  working  cylinder  G,  Fig.  105.     This 
removes  the  necessity  for  a  loose-fitting  plunger;  in  double-acting  engines,  each 
end  of  the  cylinder  is  connected  with  the  hot  (lower)  side  of  the  one  plunger  and 
with  the  cold  (upper)  side  of  the  other.     In  other  forms,  the  regenerator  has  been 
a  separate  vessel ;  in  still  others,  the  displacer  plunger  itself  became  the  regen- 
erator, being  perforated  at  the  top  and  bottom  and  filled  with  wire  gauze.     The 
Laubereau-Schwartzkopff  engine  (1)  is  identical  in  principle  with  the  Stirling, 
excepting  that  the  regenerator  is  omitted. 

The  maintenance  of  high  minimum  pressure,  as  described  in  Art.  260,  while 
not  necessarily  affecting  the  efficiency,  greatly  increases  the  capacity,  and  (since 
friction  losses  are  practically  constant)  the  mechanical  efficiency  as  well. 


A    i 


\ 


FIG.  106.     Arts.  264,  265,  2G7.  —  Stirling  Cycle. 

264.    Pressure-Volume  Diagram.     The  cycle  of  operations  described  in 
Art.  261  is  that  of  Fig.  106,  ABCD.     Considering  the  cushion  air,  the 


THE  STIRLING   ENGINE 


139 


actual  diagram  which  would  be  obtained  by  measuring  the  pressures  and 
volumes  is  quite  different.  Assume,  for  example,  that  the  total  volume 
of  cushion  air  at  maximum  pressure  (when  E  is  at  the  top  of  its  stroke 
and  H  is  just  beginning  to  move)  is  represented  by  the  distance  NE. 
Then  if  AT  be  laid  off  equal  to  NE,  the  total  volume  of  air  present  is  NI. 
Draw  an  isothermal  EFHG,  representing  the  path  of  the  cushion  air,  sep- 
arately considered,  while  the  temperature  remains  constant.  Add  its  vol- 
umes, PF,  ZH,  QG,  to  those  of  working  air,  by  laying  off  BK=  PF, 
D3f=ZH,  CL=QG,  at  various  points  along  the  stroke.  Then  the 
cycle  IKLM  is  that  actually  experienced  by  the  total  air,  assuming  the 
cushion  air  to  remain  at  constant  temperature  throughout  (Art.  262). 

The  actual  indicator  diagrams  obtained  in  tests  are  roughly  similar  to  the 
cycle  IKLM,  Fig.  106;  but  the  corners  are  rounded,  and  other  distortions  may 
appear  on  account  of  non-conformity  with  the  ideal  paths,  sluggish  valve  action, 
errors  of  the  indicating  instrument,  and  various  other  causes. 


265.    Efficiency.     The  heat  absorbed  from  the  source  along  AB,  Fig. 

-ry- 

106,    is   PA  I7^  loge  — ^  •      That    rejected    to    the  .  condenser   along    CD  is 

T7"  "^ 

PDVD\oge — £•     The  work  done  is  the  difference  of  these  two  quantities, 
^       and  the  efficiency  is 


T-t 


'  A. 


that  of  the  Carnot  cycle.     Losses  through  the  regenerator  and  by  imper- 
fection of  cycle  reduce  this  in  prac- 
tice. 


266.  Entropy  Diagram.  This  is 
given  in  Fig.  108.  T  and  t  are  the 
limiting  isotherrnals,  DA  and  BC 
the  constant  volume  curves,  along 
each  of  which  the  increase  of  en- 
tropy is  n  =  1  loge(T-M),  /  being  the 
specific  heat  at  constant  volume. 
The  gain  of  entropy  along  the  iso- 
thermals  is  obtained  as  in  Art.  253.  Ignoring  the  heat  areas  EDAFsmd. 
GCBH,  the  efficiency  is  ABCD  -*-  FABH,  that  of  the  Carnot  ^cycle.  The 
Stirling  cycle  appears  in  the  PV  diagram  of  Fig.  104  as  dkbl. 


E 

FIG.   108.      Art.  266.  — Stirling  Cycle, 
Entropy  Diagram. 


140 


APPLIED  THERMODYNAMICS 


267.   Importance  of  the  Regenerator.    Without  the  regenerator,  the  non- 
reversible  Stirling  cycle  would  have  an  efficiency  of 


This  is  readily  computed  to  be  far  below  that  of  the  corresponding  Carnot 
cycle.  The  advantage  of  the  regenerative  cycle  lies  in  the  utilization  of 
the  heat  rejected  along  BC,  Fig.  106,  thus  cancelling  that  item  in  the 
analysis  of  the  cycle.  Another  way  of  utilizing  this  heat  is  to  be 
described ;  but  while  practical  difficulties,  probably  insurmountable,  limit 
progress  in  the  application  of  the  air  engine  on  a  commercial  scale,  the 
regenerator,  upon  which  has  been  founded  our  modern  metallurgical  in- 
dustries as  well,  has  offered  the  first  possible  method  for  the  realization 
of  the  ideal  efficiency  of  Carnot  (2). 

268.  Trials.     As  early  as  1847,  a  50-hp.  Stirling  engine,  tested  at  the  Dun- 
dee Foundries,  was  shown  to  operate  at  a  thermal  efficiency  of  30  per  cent,  esti- 
mated to  be  equivalent,  considering  the  rather  low  furnace  efficiency,  to  a  coal  con- 
sumption of  1.7  Ib.  per  hp.-hr.    This  latter  result  is  not  often  surpassed  by  the  aver- 
age steam  engines  of  the  present  day.     The  friction  losses  in  the  mechanism  were 
only  11  per  cent  (3).     A  test  quoted  by  Peabody  (4)  gives  a  coal  rate  of  1.66  Ib., 
but  with  a  friction  loss  much  greater,  —  about  30  per  cent.     There  is  no  question 
as  to  the  high  efficiency  pf  the  regenerative  air  engine. 

269.  Ericsson's  Hot-air  Engine.     In  1833,  Ericsson  constructed  an  unsuccess- 
ful hot-air  engine  in  London.     About  1855,  he  built  the  steamer  Ericsson,  of  2200 
tons,  driven  by  four  immense  hot-air  engines.     After  the  abandonment  of  this 
experiment,  the  same  designer  in  1875  introduced  a  third  type  of  engine,  and  more 
recently  still,  a  small  pumping  engine,  which  has  been  extensively  applied. 

The  principle  of  the  engine  of 
1855  is  illustrated  in  Fig.  109.  B  is 
the  receiver,  A  the  displacer,  H  the 
furnace.  The  displacer  A  fits  loosely 
in  B  excepting  near  its  upper  portion, 

Jk — i.ii '..'.. , — i. .H j  where    tight   contact  is  insured  by 

if  L^iiirrSS:^  11  means  of  packing  rings.     The  lower 

!&  GS  B  b||P  portion  of  A    is    hollow,   and   filled 

B/  r^>— •'  "*^J^.  with   a  non-conductor.      The   holes 

aa   admit  air  to  the   upper   surface 
of  A.     D  is  the  compressing  pump, 

with   piston  C,  which  is  connected 

FIG.  109.    Arts.  2ti9,  270,  275.— Ericsson  Engine,     with  A  by  the  rods  dd.     E  is  a  pis- 
ton    rod    through    which    the    de- 
veloped   power    is   externally  applied.      Air  enters  the  space  above  C  through 
the  check  valve  c,  and  is  compressed  during  the  up  stroke  into  the  magazine  F 


ERICSSON   ENGINE 


141 


FIG. 


through  the  second  check  valve  e.  G  is  the  regenerator,  made  up  of  wire  gauze. 
The  control  valves,  worked  from  the  engine  mechanism,  are  at  b  and  /.  When 
b  is  opened,  air  passes  from  F  through  G  to  B,  raising  A.  Closing  of  b  at  part 
completion  of  the  stroke  causes  the  air  to  work  expansively  for  the  remainder  of 
the  stroke.  During  the  return  stroke  of  A,  air  passes  through  G,  /,  and  g  to  the 
atmosphere. 

270.  Graphical  Illustration.  The  PV  diagram  is  given  in  Fig.  110.  EBCF 
is  the  net  work  diagram,  ABCD  being  the  diagram  of  the  engine  cylinder,  AEFD 
that  of  the  pump  cylinder.  Beginning  with  A  in  its  lowest  position,  the  state  point 
in  Fig.  110  is,  for  the  engine  (lower  side  of  A),  at 
A,  and  for  the  pump  (upper  side  of  C),  at  F. 
During  about  half  the  up  stroke,  the  path  in  the 
engine  is  AB,  air  passing  to  B  from  the  re- 
generator through  5,  and  being  kept  at  constant 
pressure  by  the  heat  from  the  furnace.  During 
the  second  half  of  this  stroke,  the  supply  of  air 
from  the  regenerator  ceases,  and  the  pressure  falls 
rapidly  as  expansion  occurs,  but  the  heat  im- 
parted from  the  furnace  keeps  the  temperature 
practically  constant,  giving  the  isothermal  path 
BC.  Meanwhile,  the  pump,  receiving  air  at  the 

pressure  of  the  atmosphere,  has  been  first  compressing  it  isothermally,  or  as 
nearly  so  as  the  limited  amount  of  cooling  surface  will  permit,  along  FE,  and 
then  discharging  it  through  e  at  constant  pressure,  along  EA,  to  the  receiver  F. 
On  the  down  stroke,  the  engine  steadily  expels  the  air,  now  expanded  down  to 
atmospheric  pressure,  along  the  constant  pressure  line  CD,  while  the  pump  simi- 
larly draws  in  air  from  the  atmosphere  at  constant  pressure  along  DF.  At  the  end 
of  this  stroke,  the  air  in  F,  at  the  state  A,  is  admitted  to  the  engine.  The  ratio  of 

pump  volume  to  engine  volume  is  FD  -=-  DC,  or  — • 

271.  Efficiency.  The  Ericsson  cycle  be- 
longs to  the  same  class  as  that  of  Stirling, 
being  bounded  by  two  isothermals  and  two 
like  polytropics ;  but  the  polytropics  are  in 
this  case  constant  pressure  lines  instead  of 
constant  volume  lines.  The  net  entropy 
diagram  EBCF,  Fig.  Ill,  is  similar  to  that 
of  the  Stirling  engine,  but  the  isodiabatics 
swerve  more  to  the  right,  since  k  exceeds  I, 


110.      Arts.  270,  272,  273.— 
Ericsson  Cycle. 


FIG.  111.  Art.  271.— Ericsson  Cycle, 
Entropy  Diagram. 


while  the  efficiency  is  the  same  as  that  of  the  Stirling  engine, 


T-t 


272.  Tests.  As  computed  by  Rankine  from  Norton's  tests,  the  effi- 
ciency of  the  steamer  Ericsson's  engines  was  26.3  per  cent ;  the  efficiency 
of  the  furnace  was,  however,  only  40  per  cent.  The  average  effective  pres- 


142  APPLIED  THERMODYNAMICS 

sure  (EBCF+XC,  Fig.  110)  was  only  2.12  Ib.  The  friction  losses  were 
enormous.  A  small  engine  of  this  type  tested  by  the  writer  gave  a  con- 
sumption of  15.64  cu.  ft.  of  gas  (652  B.  t.  u.  per  cubic  foot)  per  Ihp.-hr. ; 
equivalent  to  170  B.  t.  u.  per  Ihp.-minute;  and  since  1  horse  power 
=  33,000  foot-pounds  =  33,000  -f-  778  =  42.45  B.  t.  u.  per  minute,  the 
thermodynamic  efficiency  of  the  engine  was  42.45  -*-  170  =  0.25. 

273.  Actual  Designs.  In  order  that  the  lines  FC  and  EB,  Fig.  110*  may  be 
horizontal,  the  engine  should  be  triple  or  quadruple,  as  in  the  steamer  Ericsson,  in 
which  each  of  the  four  cylinders  had  its  own  compressing  pump,  but  all  were  con- 
nected with  the  same  receiver,  and  with  a  single  crank  shaft  at  intervals  of  a 
quarter  of  a  revolution.  Specimen  indicator  diagrams  are  given  in  Figs.  107,  112. 


FIG.   107.     Art.   273.— Indicator  FIG.  112.    Art.  273.— Indicator 

Card  from  Ericsson  Engine.  Diagram,  Ericsson  Engine. 

274.  Testing  Hot-air  Engines.     It  is  difficult  to  directly  and  accurately  meas- 
ure the  limiting  temperatures  in  an  air  engine  test,  so  that  a  comparison  of  the 
actually  attained  with  the  computed  ideal  efficiencies  cannot  ordinarily  be  made. 
Actual  tests  involve  the  measurement  of  the  fuel  supplied,  determination  of  its 
heating  value,   and  of  the  indicated    and  effective   horse  power   of  the   engine 
(Art.  487).     These  data  permit  of  computation  of  the  thermal   and  mechanical 
efficiencies,  the  latter  being  of  much  importance.     In  small  units,  it  is  sometimes 
as  low  as  0.50. 

• 

275.  The  Air  Engine  as  a  Heat  Motor.    In  nearly  every  large  application,  the 
hot-air  engine  has  been  abandoned  on  account  of  the  rapid  burning  out  of  the 
heating  surfaces  due  to  their  necessarily  high  temperature.     Napier  and  Rankine 
(5)  proposed  an  "  air  heater,"  designed  to  increase  the  transmissive  efficiency  of 
the  heating  surface.     Modern  forms  of  the  Stirling  or  Ericsson  engines,  in  small 
units,  are  comparatively  free  from  this  ground  of  objection.     Their  design  permits 
of  such  amounts  of  heat- transmitting  surface  as  to  give  grounds  for  expecting  a 
much  less  rapid  destruction  of  these  parts.     It  has  been  suggested  that  excessive 
bulk  may  be  overcome  by  using  higher  pressures.     (Zeuner  remarks  (6)  that  the 
bulk  is  not  excessive  when  compared  with  that  of  a  steam  engine  with  its  auxiliary 
boiler  and  furnace).     Rankine  has  suggested  the  introduction  of  a  second  com- 
pressed air  receiver,  in  Fig.  109,  from  which  the  supply  of  air  would  be  drawn 
through  c,  and  to  which  air  would  be  discharged  through  f.     This  would  make  the 
engine  a  "  closed  "  engine,  in  which  the  minimum  pressure  could  be  kept  fairly 
high;  a  small  air  pump  would  be  required  to  compensate  for  leakage.     A  ''con- 
denser "  would  be  needed  to  supplement  the  action  of  the  regenerator  by  more 


HOT-AIR   ENGINES  143 

thoroughly  cooling  the  discharged  air,  else  the  introduction  of  "  back  pressure  " 
would  reduce  the  working  range  of  temperatures.  The  loss  of  the  air  by  leakage, 
and  consequent  waste  of  power,  would  of  course  increase  with  increasing  pressures. 
Instead  of  applying  heat  externally,  as  proposed  by  Joule,  in  the  engine  shown 
in  Fig.  101,  there  is  no  reason  why  the  combustion  of  the  fuel  might  not  proceed 
within  the  hot  chamber  itself,  the  necessary  air  for  combustion  being  supplied  by 
the  pump.  The  difficulties  arising  from  the  slow  transmission  of  heat  would  thus 
be  avoided.  An  early  example  of  such  an  engine  applied  in  actual  practice  was 
Cayley's  (7),  later  revived  by  Wenham  (8)  and  Buckett  (9).  In  such  engines, 
the  working  fluid,  upon  the  completion  of  its  cycle,  is  discharged  to  the  atmos- 
phere. The  lower  limit  of  pressure  is  therefore  somewhat  high,  and  for  efficiency 
the  necessary  wide  range  of  temperatures  involves  a  high  initial  pressure  in  the 
cylinder.  The  internal  combustion  air  engine  even  in  these  crude  forms  may  be 
regarded  as  the  forerunner  of  the  modern  gas  engine. 

(1)  Zeuner,  Technical  Thermodynamics  (Klein),  1907,  I,  340.  (2)  The  theoreti- 
cal basis  of  regenerator  design  appears  to  have  been  treated  solely  by  Zeuner,  op.  cit,, 
I,  314-323.  (3)  Rankine,  The  Steam  Engine,  1897,  368.  (4)  Thermodynamics  of  the 
Steam  Engine,  1907,  302.  (5)  The  Steam  Engine,  1897,  370.  (6)  Op.  cit.,  I,  381. 
(7)  Nicholson's  Art  Journal,  1807  ;  Min.  Proc.  Inst.  C.  E.,  IX.  (8)  Proc.  Inst. 
Mech.  Eng.,  1873.  (9)  Inst.  Civ.  Eng.,  Heat  Lectures,  1883-1884;  Min.  Proc.  Inst. 
C.  E.,  1845,  1854. 


SYNOPSIS  OF   CHAPTER   X 

The  hot-air  engine  proper  is  an  external  combustion  motor  of  the  open  or  closed  type. 
The  temperature  of  a  permanent  gas  may  be  varied  independently  of  the  pressure ;  this 
makes  the  possible  efficiency  higher  than  that  attainable  in  vapor  engines. 

\  —  (  —  \"     •  the  Carnot  cycle  leads  to  either  excessive  pressures  or  an  enormous 
PJ      \tJ 

cylinder. 

The  polytropic  cycle  is  bounded  by  two  pairs  of  isodiabatics. 

The  Lorenz  cycle  is  bounded  by  a  pair  of  adiabatics  and  a  pair  of  unlike  polytropics. 
The  Beitlinyer  (isodiabatic)  cycle  is  bounded  by  a  pair  of  isothermals  and  a  pair  of 

isodiabatics. 
The  Joule  engine  works  in  a  cycle  bounded  by  two  constant  pressure  lines  and  two 

adiabatics  ;  its  efficiency  is  T°~  t . 

T0 

The  regenerator  is  a  "fly  wheel  for  heat."  Any  cycle  bounded  by  a  pair  of  iso- 
thermals and  a  pair  of  like  polytropics,  if  worked  with  a  regenerator,  has  an  ideal 
efficiency  equal  to  that  of  the  Carnot  cycle  ;  the  heat  rejected  along  one  polytropic 
is  absorbed  by  the  regenerator,  which  in  turn  emits  it  along  the  other  polytropic, 
the  operation  being  subject  to  slight  losses  in  practice. 

The  Stirling  cycle,  bounded  by  a  pair  of  isothermals  and  a  pair  of  constant  volume 
curves  :  correction  of  the  ideal  PV  diagram  for  cushion  air  :  comparison  with  indi- 
cator card  ;  the  entropy  diagram  ;  efficiency,  formulas  with  and  without  the  regen- 
erator ;  coal  consumption,  1.7  Ib.  per  hp.-hr. 

The  Ericsson  cycle,  bounded  by  a  pair  of  isothermals  and  a  pair  of  constant  pressure 
curves  :  efficiency  from-  fuel  to  power,  26  per  cent. 


144  APPLIED  THERMODYNAMICS 

By  designing  as  "closed"  engines,  the   minimum  pressure  may  be  raised  and  the 

capacity  of  the  cylinder  increased. 
The  air  engine  is  unsatisfactory  in  large  sizes  on  account  of  the  rapid  burning  out  of 

the  heating  surfaces  and  the  small  capacity  for  a  given  bulk. 


PROBLEMS 

(NOTE.   Considerable  accuracy  in  computation  will  be  found  necessary  in  solving  Prob- 
lems 4  a  and  5). 

1.  How  much  greater  is  the  ideal  efficiency  of  an  air  engine  working  between  tem- 
perature limits  of  2900°  F.  and  600°  F.  than  that  of  the  steam  engine  described  in  Prob- 
lem 5,  Chapter  VI  ? 

2.  Plot  to  scale  (1  inch  =  2  cu.  ft.  =  40  Ib.  per  square  inch)  the  PV  Carnot  cycle 
for  T=  600°,  t  =  500°  (both  absolute)  the  lowest  pressure  being  14.7  Ib.  per  square 
inch,  the  substance  being  one  pound  of  air,  and  the  volume  ratio  during  isothermal 
expansion  being  12.6. 

3.  In  Problem  2,  if  the  upper  isothermal  be  made  700°  absolute,  what  will  be  the 
maximum  pressure  ? 

4  a.  Plot  the  entropy  diagram,  and  find  the  efficiency,  of  a  polytropic  cycle  for  air 
between  600°  F.  and  500°  F.,  in  which  n  =  1.3,  m  =  -  1.3,  the  pressure  at  d  (Fig.  95) 
is  18  Ib.  per  square  inch,  and  the  pressure  at  e  (Fig.  95)  is  22  Ib.  per  square  inch. 

4  6.    In  Art.  251,  prove  that  7> :  Tb  : :  Td  :  Te,  and  also  that  Pd  :  Pe : :  Pf :  P5. 

5.  Plot  the  entropy  diagram,  and  find  the  efficiency,  of  a  Lorenz  cycle  for  air 
between  600°  F.  and  500°  F.,  in  which  n  =  —  1.3,  q  =  0.4,  the  highest  pressure  being 
60  Ib.  per  square  inch  and  the  temperature  at  gr,  Fig.  97,  being  550°  F. 

6.  Plot  the  entropy  diagram,  and  find  the  efficiency,  of  a  Reitlinger  cycle  between 
600°  F.  and  500°  F.,  when  n  =  1.3,  the  maximum  pressure  is  80  Ib.  per  square  inch,  the 
ratio  of  volumes  during  isothermal  expansion  12,  and  the  working  substance  one 
pound  of  air. 

7.  Show  that  in  the  Joule  engine  the  efficiency  is   T~jTq,  Art.  255. 

8.  Plot  the  entropy  diagram,  and  find  the  efficiency,  of  a  Joule  air  engine  working 
between  600°  F.  and  -  200°  F.,  the  maximum  pressure  being  100  Ib.  per  square  inch, 
the  ratio  of  volumes  during  adiabatic  expansion  2,  and  the  weight  of  substance  2  Ib. 

9.  Plot  PV  and  NT  diagrams  for  one  pound  of  air  worked  between  3000°  F.  and 
400°  F. :  (a)  in  the  Carnot  cycle,  (6)  in  the  Ericsson  cycle,  (c)  in  the  Stirling  cycle,  the 
extreme  pressure  range  being  from  50  to  2000  Ib.  per  square  inch. 

10.    Find  the  efficiencies  of  the  various  cycles  in  Problem  9,  without  regenerators. 
'  11.    Compare  the  efficiencies  in  Problems  4  a,  5,  and  6,  with  that  of  the  correspond- 
ing Carnot  cycle. 

12.  An  air  engine  cylinder  working  in  the  Stirling  cycle  between  1000°  F.  and 
2000°  F.,  with  a  regenerator,  has  a  volume  of  1  cu.  ft.     The  ratio  of  expansion  is  3. 
By  what  percentages  will  the  capacity  and  efficiency  be  affected  if  the  lower  limit  of 
pressure  is  raised  from  14.7  to  85  Ib.  per  square  inch  ? 

13.  In  the  preceding  problem,  one  eighth  of  the  cylinder  contents  is  cushion  air,  at 
1000°  F.    Plot  the  ideal  indicator  diagram  for  the  lower  of  the  two  pressure  limits,  cor- 
rected for  cushion  air. 


HOT-AIR  ENGINES  145 

14.  In  Art.  268,  assuming  that  the  coal  used  in  the  Dundee  foundries  contained 
14,000  B.  t.  u.  per  pound,  what  was  the  probable  furnace  efficiency?    In  the  Peabody 
test,  if  the  furnace  efficiency  was  80  per  cent,  and  the  coal  contained  14,000  B.  t.  u., 
what  was  the  thermal  efficiency  of  the  engine  ? 

15.  What  was  the  efficiency  of  the  plant  in  the  steamer  Ericsson  ? 

16.  Sketch  the  TWand  PV  diagrams,  within  the  same  temperature  and  entropy 
limits,  of  all  of  the  cycles  discussed  in  this  chapter,  with  the  exception  of  that  of  Joule. 
Why  cannot  the  Joule  and  Ericsson  cycles  be  drawn  between  the  same  limits  ?     Show 
graphically  that  in  no  case  does  the  efficiency  equal  that  of  the  Carnot  cycle. 

17.  Compare  the  cycle  areas  in  Problem  9. 

18.  In  Problem  2,  what  is  the  minimum  possible  range  of  pressures  compatible 
with  a  finite  work  area  ?    Illustrate  graphically. 

19.  Derive  a  definite  formula  for  the  efficiency  of  the  Reitlinger  cycle,  Art.  253. 


CHAPTER  XI 

GAS  POWER 

THE  GAS  PRODUCER 

276.  History.  The  bibliography  (1)  of  internal  combustion  engines  is  exten- 
sive, although  their  commercial  development  is  of  recent  date.  Coal  gas  was  dis- 
tilled as  early  as  1691 ;  the  waste  gases  from  blast  furnaces  were  first  used  for 
heating  in  1809.  The  first  English  patent  for  a  gas  engine  approaching  modern 
form  was  granted  in  1794.  The  advantage  of  compression  was  suggested  as  early 
as  1801,  but  was  not  made  the  subject  of  patent  until  1838  in  England  and  1861  in 
France.  Lenoir,  in  1860,  built  the  first  practical  gas  engine,  which  developed  a 
thermal  efficiency  of  0.04.  The  now  familiar  polytropic  "  Otto  "  cycle  was  pro- 
posed by  Beau  de  Rochas  at  about  this  date.  The  same  inventor  called  attention 
to  the  necessity  of  high  compression  pressures  in  1862 ;  a  principle  applied  in 
practice  by  Otto  in  1874.  Meanwhile,  in  1870,  the  first  oil  engine  had  been  built. 
The  four-cycle  cornpressive  Otto  "silent"  engine  was  brought  out  in  1876,  show- 
ing a  thermal  efficiency  of  0.15,  a  result  better  than  that  then  obtained  in  the  best 
steam  power  plants. 

If  the  isothermal,  isometric,  isopiestic,  and  adiabatic  paths  alone  are  considered, 
there  are  possible  at  least  twenty-six  different  gas  engine  cycles  (2).  Only  four 
of  these  have  had  extended  development ;  of  these  four,  only  two  have  survived. 
The  Lenoir  (3)  and  Hugon  (4)  non-compressive  engines  are  now  represented  only 
by  the  Bischoff  (5).  The  Barsanti  "free  piston"  engine,  although  copied  by 
Gilles  and  by  Otto  and  Laugen  (1856)  (6),  is  wholly  obsolete.  The  variable  vol- 
ume engine  of  Atkinson  (7)  was  commercially  unsuccessful. 

Up  to  1885,  illuminating  gas  was  commonly  employed,  only  small  engines 
were  constructed,  and  the  high  cost  of  the  gas  prevented  them  from  being  com- 
mercially economical.  Nevertheless,  six  forms  were  exhibited  in  1887.  The 
Priestman  oil  engine  was  built  in  1888.  With  the  advent  of  the  Dowson  process, 
in  1878,  with  its  possibilities  of  cheap  gas,  advancement  became  rapid.  By  1897, 
a  400-hp.  four-cylinder  engine  was  in  use  on  gas  made  from  anthracite  coal.  At 
the  present  time,  double-acting  engines  of  5400  hp.  have  been  placed  in  operation ; 
still  larger  units  have  been  designed,  and  a  few  applications  of  gas  power  have 
been  made  even  in  marine  service. 

Natural  gas  is  now  transmitted  to  a  distance  of  200  miles,  under  300  Ib.  pres- 
sure. Illuminating  gas  has  been  pumped  52  miles.  Martin  (8)  has  computed  that 
coal  gas  might  be  transmitted  from  the  British  coal  fields  to  London  at  a  delivered 
cost  of  15  cents  per  1000  cu.  ft.  His  plan  calls  for  a  25-inch  pipe  line,  at  500  Ib. 
initial  pressure  and  250  Ib.  terminal  pressure,  carrying  40,000,000,000  cu.  ft.  of 

146 


GAS   POWER  147 

gas  per  year.     The  estimated  46,000  hp.  required  for  compression  would  be  derived 
from  the  waste  heat  of  the  gas  leaving  the  retorts. 

Producer  gas  is  even  more  applicable  to  heating  operations  than  for  power 
production.  It  is  meeting  with  extended  use  in  ceramic  kilns  and  for  ore  roast- 
ing, and  occasionally  even  for  firing  steam  boilers. 

277.  The  Gas  Engine   Method.     The   expression   for  ideal  efficiency, 
(T—  t)-s-T,  increases  as  T  increases.    In  a  steam  plant,  although  boiler  fur- 
nace temperatures  of  2500°  F.  or  higher  are  common,  the  steam  passes  to 
the  engine,  ordinarily,  at  not  over  350°  F.     This  temperature  expressed  in 
absolute  degrees  limits  steam  engine  efficiency.     To  increase  the  value  of 
Tj  either  very  high  pressure  or  superheat  is  necessary,  and  the  practicable 
amount  of  increase  is  limited  by  considerations  of  mechanical  fitness  to 
withstand  the  imposed  pressures  or  temperatures.     In  the  internal  com- 
bustion engine,  the  working  substance  reaches  a  temperature  approximat- 
ing 3000°  F.  in  the  cylinder.     The  gas  engine  has  therefore  the  same  ad- 
vantage as  the  hot  air  engine,  —  a  wide  range  of  temperature.     Its  working 
substance  is,  in  fact,  for  the  most  part  heated  air.     The  fuel,  which  may 
be  gaseous,  liquid,  or  even  solid,  is  injected  with  a  proper  amount  of  air, 
and  combustion  occurs  within  the  cylinder.     The  disadvantage  of  the  ordi- 
nary hot  air  engine  has  been  shown  to  arise  from  the  difficulty  of  trans- 
mitting heat  from  the  furnace  to  the  working  substance.     In  this  respect, 
the  gas  engine  has  the  same  advantage  as  the  steam  engine,  —  large  capa- 
city for  its  bulk,  —  for  there  is  no  transmission  of  heat ;  the  cylinder  is 
the  furnace,  and  the  products  of  combustion  constitute  the  working  sub- 
stance.    A  high  temperature  of  working  substance  is  thus  possible,  with 
large  work  areas  on  the  pv  diagram,  and  a  rapid  rate  of  heat  propagation. 

In  the  gas  engine,  then,  certain  chemical  changes  which  constitute  the  pro- 
cess described  as  combustion,  must  be  considered ;  although  such  changes  are  in  gen- 
eral not  to  be  included  in  the  phenomena  of  engineering  thermodynamics. 

278.  Fuels.     The  common  fuels  are  gases  or  oils.     In  some  sections,  natural 
gas  is  available.     This   is  high  in  heating  value,  consisting  mainly  of  methane, 
CH4.     Carbureted  water  gas,  used  for  illumination,  is  nearly  as  high  in  heating 
value,  consisting  of  approximately  equal  volumes  of  hydrogen,  carbon  monoxide, 
and  methane,  with  some  ethylene  and  traces  of  other  substances.     Uncarbureted 
(blue)  water  gas  is  almost  wholly  carbon  monoxide  and  hydrogen.     Its  heating 
value  is  less  than  half  that  of  the  carbureted  gas.     Both  water  gas  and  coal  gas 
are  uneconomical  for  power  production ;  in  the  processes  of  manufacture,  large 
quantities  of  coal  are  left  behind  as  coke.     Coal  gas,  consisting  principally  of  hy- 
drogen and  methane,   is  slightly  lower  in  heating  value  than  carbureted  water 
gas.     It  is  made  by  distilling  soft  coal  in  retorts,  about  two  thirds  of  the  weight 
of  coal  becoming  coke.     Coke  oven  gas  is  practically  the  same  product;  the  main 
output  in  its  case  being  coke,  while  in  the  former  it  is  gas. 


148  APPLIED  THERMODYNAMICS 

Producer  gas  ("  Dowson  "  gas,  "  Mond  "  gas,  etc.)  is  formed  by  the  par- 
tial combustion  of  coal  in  air.  It  is  essentially  carbon  monoxide,  diluted 
with  large  quantities  of  nitrogen  and  consequently  low  in  heating  value. 
Its  exact  composition  varies  according  to  the  fuel  from  which  it  is  made, 
the  quantity  of  air  supplied,  etc.  When  soft  coal  is  used,  or  when  much 
steam  is  fed  to  the  producer,  large  proportions  of  hydrogen  are  present. 

It  is  of  no  value  as  an  illuminant.  Blast  furnace  gas  is  producer  gas 
obtained  as  a  by-product  on  a  large  scale  in  metallurgical  operations.  It  contains 
less  hydrogen  than  ordinary  producer  gases,  since  steam  is  not  employed  in  its 
manufacture,  and  is  generally  quite  variable  in  its  composition  on  account  of  the 
exigencies  of  furnace  operation.  Acetylene,  C2H2,  is  made  by  combining  calcium 
carbide  and  water.  It  has  an  extremely  high  heating  and  illuminating  value. 
All  hydrocarbon  aceous  substances  maybe  gasified  by  heating  in  closed  vessels; 
gases  have  in  this  way  been  produced  from  peat,  sawdust,  tan  bark,  wood,  garbage, 
animal  fats,  etc. 

279.  Oil  Gases.     Many  liquid  hydrocarbons  may  be  vaporized  by  appropriate 
methods,  under  conditions  which  make  them  available  for  gas  engine  use.     Some 
of  these  liquids  must  be  vaporized  by  artificial  heat  and  then  immediately  used,  or 
they  will  again  liquefy  as  their  temperatures  fall.     The  vaporizer  or  "  carburetor  " 
is  therefore  located  at  the  engine,  where  it  atomizes  each  charge  of  fuel  as  required. 
Gasoline  is  most  commonly  used ;  its  vapor  has  a  high  heating  value.     Kerosene, 
and,  more  recently,  alcohol,  have  been  employed.     By  mixing  gasoline  and  air  in 
suitable  proportions,  a  saturated  or  "  carbureted  "  air  is  produced.     This  acts  as 
a  true  gas,  and  must  be  mixed  with  more  air  to  permit  of  combustion.     A  gas 
formed  in  the  proportion  of  1000  cu.  ft.  of  air  to  2  gallons  of  liquid  gasoline,  for 
example,  does  not  liquefy.     A  third  form  of  oil  gas  is  produced  by  heating  certain 
hydrocarbons  without   air ;   the   "  cracking "   process  produces,  first,   less  dense 
liquids,  and,  finally,  gaseous  bodies,  which  do  not  condense.     The  process  must  be 
carried  on  in  a  closed  retort,  and  arrangements  must  be  made  for  the  removal  of 
residual  tar  and  coke. 

280.  Liquid  Fuels.     These  have  advantages  over  solid  or  gaseous  fuels,  aris- 
ing from  the  usually  large  heating  value  per  unit  of  bulk,  and  from  ease  of  trans- 
portation.    All  animal  and  vegetable  oils  and  fats  may  be  reduced  to  liquid  fuels; 
those  oils  most  commonly  employed,  however,  are  petroleum  products.     Crude 
petroleum  may  be  used ;  it  is  more  customary  to  transform  this  to  "  fuel  oil "  by 
removing  the  moisture,  sulphur,  and  sediment;  and  some  of  these  "fuel  oils"  are 
used  in  gas  engines.     Of  petroleum  distillates,  the  gasolines  are  most  commonly 
utilized  in  this  country.     They  include  an  86°  liquid,  too  dangerous  for  commer- 
cial purposes;  the  74°  "benzine,"  and  the  69°  naphtha.     "Distillate,"  an  impure 
kerosene,  from  which  the  gasoline  has  not  been   removed,  is  occasionally  used. 
Both  grain  alcohol  (C2H6O)  and  wood  alcohol  (CH4O)  have  been  used  in  gas  en- 
gines (9).     Various  distillates  from  brown  and  hard  coal  tars  have  been  employed 
in  Germany.     Their  suitability  for  power  purposes  varies  with  different  types  of 
engines.     The  benzol  derived  from  coal  gas  tar  has  been  successfully  used ;  the 
brown  coal  series,  CnH2,4,  CnH2n+2,  CnH2n_2,  contains  many  useful  members  (10). 


THE  GAS  PRODUCER 


149 


281.  The  Gas  Producer.  This  essential  auxiliary  of  the  modern  gas 
engine  is  made  in  a  large  number  of  types,  one  of  which  is  shown  in  Fig. 
113.  This  is  a  brick-lined  cylindrical  shell,  set  over  a  water-sealed  pit  P, 
on  which  the  ash  bed  rests.  Air  is  forced  in  by  means  of  the  steam  jet 
blower  A,  being  distributed  by  means  of  the  conical  hood  B,  from  which 


\ 

E=^J 

<4s$xd 

^ 

SL 

p  JS 

aprorarajS^^ 

5$sV// 
| 

1 

.'"  ~~  •' 

*  °'---^;V 

§ 

FIG.  113.    Art.  281.  — The  Amsler  Gas  Producer. 


it  passes  up  to  the  red-hot  coal  bed  above.  Here  carbon  dioxide  is  formed 
and  the  steam  decomposes  into  hydrogen  and  oxygen.  Above  this  "  com- 
bustion zone"  extends  a  layer  of  coal  less  highly  heated.  The  carbon 
dioxide,  passing  upward,  is  decomposed  to  carbon  monoxide  and  oxygen. 
The  hot  mixed  gases  now  pass  through  the  freshly  fired  coal  at  the  top  of 
the  producer,  causing  the  volatile  hydrocarbons  to  distill  off,  the  entire 
product  passing  out  at  (7.  The  coal  is  fed  in  through  the  sealed  hopper  D. 


150  APPLIED  THERMODYNAMICS 

At  E  are  openings  for  the  bars  used  to  agitate  the  fire.     At  F  are  peep- 
holes. 

An  automatic  feeding  device  is  sometimes  used  at  D.  The  air  may 
be  forced  in  by  a  blower,  or  sucked  through  by  an  exhauster,  or  by  the 
engine  piston  itself,  displacing  the  steam  jet  blower  A.  The  fuel  may 
be  supported  on  a  solid  grate,  or  on  the  bottom  of  a  producer  without  the 
water  seal;  grates  may  be  either  stationary  or  mechanically  operated. 
Mechanical  agitation  may  be  employed  instead  of  the  poker  bars  inserted 
through  E.  Sometimes  water  gas,  for  illumination,  and  producer  gas,  for 
power,  are  made  in  the  same  plant.  Two  producers  are  then  employed, 
the  air  blast  being  applied  to  one,  while  steam  is  decomposed  in  the  other. 

Provision  must  be  made  for  purifying  the  gas,  by  deflectors,  wet  and  dry 
scrubbers,  niters,  coolers,  etc.  For  the  removal  of  tar,  which  would  be  seriously 
objectionable  in  engines,  mechanical  separation  and  washing  are  useful,  but  the 
complete  destruction  of  this  substance  involves  the  passing  of  the  gas  through  a 
highly  heated  chamber;  this  may  be  a  portion  of  the  producer  itself,  as  in 
"  under-feed,"  "  inverted  combustion,"  or  "  down-draft "  types  :  causing  the  trans- 
formation of  the  tar  to  fixed  gases.  On  account  of  the  difficulty  of  tar  removal, 
anthracite  coal  or  coke  or  semi-bituminous,  non-caking  coal  must  generally  be  used 
in  power  plants.  The  air  supplied  to  the  producer  is  sometimes  preheated  by  the 
sensible  heat  of  the  waste  gases,  in  a  "  recuperator."  The  "  regenerative  "  prin- 
ciple—  heating  the  air  and  gas  delivered  to  the  engine  by  means  of  the  heat  of 
the  exhaust  gases  —  is  inapplicable,  for  reasons  which  will  appear. 

282.  The  Producer  Plant.     The   ordinary  producer  operates  under  a  slight 
pressure ;  in  the  suction  type,  now  common  in  small  plants,  the  engine  piston 
draws  air  through  the  producer  in  accordance  with  the  load  requirements.     Pres- 
sure producers  have  been  used  on  extremely  low  grade  fuels:  Jahn,  in  Germany, 
has,  it  is  reported,  gasified  mine  waste  containing  only  20  per  cent  of  coal.     Suc- 
tion producers,  requiring  much  less  care  and  attention,  are  usually  employed  only 
on  the  better  grades  of  fuel.     Most  producers  require  a  steam  blast;  the  steam 
must  be  supplied  by  a  boiler  or  "  vaporizer,"  which  in  many  instances  is  built  as  a 
part  of  the  producer,  the  superheated, steam  being  generated  by  the  sensible  heat 
carried  away  in  the  gas.     Automatic  operation  is  effected  in  various  ways:   in 
the  Amsler  system,  by  changing  the  proportion  of  hydrogen  in  the  gas,  involving 
control  of  the  steam  supply ;  in  the  Pintsch  process,  by  varying  the  draft  at  the 
producer  by  means  of  an  inverted  bell,  under  the  control  of  a  spring,  from  beneath 
which  the  engine  draws  its  supply;  and  in  the  Wile  apparatus,  by  varying  the 
draft  by  means  of  valves  operated  from  the  holder.     Figure  114  shows  a  complete 
producer  plant,  with  separate  vaporizer,  economizer  (recuperator),  and  holder  for 
storing  the  gas  and  equalizing  the  pressure. 

283.  By-product  Recovery.     Coal  contains  from  0.5  to  3  per  cent  of  nitrogen, 
about  15  per  cent  of  which  passes  off  in  the  gas  as  ammonia.     The  successful 
development  of  the  Mond  process  has  demonstrated  the  possibility  of  recovering 
this  in  the  form  of  ammonium  sulphate,  a  valuable  fertilizing  agent. 


THE  GAS  PRODUCER 


151 


152  APPLIED  THERMODYNAMICS 

284.  Action  in  the  Producer.     Coal  is  gasified  on  the  producer  grate. 
Ideally,  this  coal  is  carbon,  and  leaves  the  producer  as  carbon  monoxide, 
4450  B.  t.  u.  per  pound  of  carbon  having  been  expended  in  gasification. 
Then  only  10,050  B.  t.  u.  per  pound  of  carbon  are  present  in  the  gas,  and 
the  efficiency  cannot  exceed  10,050  -j- 14,500  =  0.694.    The  4450  B.  t.  u.  con- 
sumed in  gasification  are  evidenced  only  in  the  temperature  of  the  gas. 
With  actual  conditions,  the  presence  of  carbon  dioxide  or  of  free  oxygen 
is  an  evidence  of  improper  operation,  further  decreasing  the  efficiency.    By 
introducing  steam,  however,  decomposition  occurs  in  the  producer,  the  tem- 
perature of  the  gas  is  reduced,  and  available  hydrogen  is  carried  to  the 
engine ;  and  this  action  is  essential  to  producer  efficiency  for  power  pur- 
poses, since  a  high  temperature  of  inlet  gas  is  a  detriment  rather  than  a 
benefit  in  engine  operation.    The  ideal  efficiency  of  the  producer  may  thus 
be  brought  up  to  something  over  80  per  cent;  a  limit  arising  when  the 
proportion  of  steam  introduced  is  such  as  to  reduce  the  temperature  of  the 
gas  below  about  1800°  F.,  when  the  rate  of  decomposition  greatly  decreases. 
The  proportion  of  steam  to  air,  by  weight,  is  then  about  6  per  cent,  the 
heating  value  of  the  gas  is  increased,  the  percentage  of  nitrogen  decreased, 
and  nearly  20  per  cent  of  the  total  oxygen  delivered  to  the  producer  has 
been  supplied  by  decomposed  steam.    A  similar  result  may  be  attained  by 
introducing  exhausted  gas  from  the  engine  to  the  producer.     The  carbon 
dioxide  in  this  gas  decomposes  to  monoxide,  which  is  carried  to  the  engine 
for  further  use.     This  method  is  practiced  in  the  Mond  system,  and  has 
had  other  applications.     To  such  extent  as  the  coal  is  hydrocarbonaceous, 
however,  the  ideal  efficiency,  irrespective  of  the  use  of  either  steam  or 
waste  gas,  is  100  per  cent.     Figure  115  shows  graphically  the  results  com- 
puted as  following  the  use  of  either  steam  or  waste  gases  with  pure  car- 
bon as  the  fuel.    The  maximum  ideal  efficiency  is  about  3J  per  cent  greater 
when  steam  is  used,  if  the  temperature  limit  is  fixed  at  1800°. F.,  but  the 
waste  gases  give  a  more  uniform  (though  less  rich)  gas.     The  higher  ini- 
tial temperature  of  the  waste  gases  puts  their  use  practically  on  a  parity 
with  that  of  steam.     Either  system  tends  to  prevent  clinkering.     The 
maximum  of  producer  efficiency,  for  power  gas  purposes,  is  ideally  from 
5  to  10  per  cent  less  than  that  of  the  steam  boiler.     High  percentages  of 
hydrogen  resulting  from  the  excessive  use  of  steam  may  render  the  gas 
too  explosive  for  safe  use  in  an  engine  (10  a)  (25). 

285.  Example  of  Computation.     Let  20  per  cent  of  the  oxygen  necessary  for 
gasifying  pure  carbon  be  supplied  by  steam.     Each  pound  of  fuel  requires  1£  Ib. 
of  oxygen  for  conversion  to  carbon  monoxide.    Of  this  amount,  0.20  x  1^  =  0.2666  Ib. 
will  then  be  supplied  by  steam;  and  the  balance,  1.0667  Ib.,  will  be  derived  from 
the  air,  bringing  in  with  it  ff  x  1.0667  =  3.57  Ib.  of  nitrogen.     The  oxygen  derived 
from  steam  will  also  carry  with  it  £  x  0.2666  =  0.0333  Ib.  of  hydrogen.     The  pro- 
duced gas  will  contain,  per  pound  of  carbon, 


PRODUCER  EFFICIENCY  153 

2.33  Ib.  carbon  monoxide, 
3.57  Ib.  nitrogen, 
0.0333  Ib.  hydrogen. 

The  heat  evolved  in  burning  to  monoxide  is  4450  B.  t.  u.  per  pound.     A  por- 
tion of  this,  however,  has  been  put  back  into  the  gas,  the  temperature  having  been 


Waste  Gas  supplied;  Percentage  of  Fuel  gasified  by  Weight  . 
109  202        256  382 


3    4     5    6    7    8    9    10   II    12    13   14   15   16   17 
Percentage  of  Steam  by  Weight. 

FIG.  115.    Art,  284.— Reactions  in  the  Producer. 


lowered  by  the  decomposition  of  the  steam.  Under  the  conditions  existing  in  the 
producer,  the  heat  of  decomposition  is  about  62,000  B.  t.  u.  per  pound  of  hydrogen. 
The  net  amount  of  heat  evolved  is  then  4450  -  (0.0333  x  62,000)  =  2383  B.  t.  u., 


154  APPLIED  THERMODYNAMICS 

and  the  efficiency  is  14'°00  ~  2888  =  0.84.    The  rise  in  temperature  is  computed  as 
14,500 

follows :  to  heat  the  gas  1°  F.  there  are  required 

WEIGHT  SPECIFIC  HEAT 

For  carbon  monoxide,         2.33        x  0.2479         =  0.578  B.  t.  u. 

For  nitrogen,  3.57        x  0.2438         =  0.869  B.  t.  u. 

For  hydrogen,  0.0333     x  3.4  =  0.113  B.  t.  u. 

a  total  of     1.560  B.  t.  u. 

The  .2383    B.    t.    u.   evolved  will   then  cause   an   elevation   of  temperature   of 

-^  =  1527°  F. 
1.560 

With  pure  air  only,  used  for  gasifying  pure  carbon,  the  gas  would  consist  of 
2^  Ib.  of  carbon  monoxide  and  4.45  Ib.  of  nitrogen  ;  the  percentages  being  34.5 
and  65.5.  For  an  actual  coal,  the  ideal  gas  composition  may  be  calculated  on  the 
assumptions  that  the  hydrogen  and  hydrocarbons  pass  off  unchanged,  and  that  the 
carbon  requires  1^  times  its  own  weight  of  oxygen,  part  of  which  is  contained  in 
the  fuel,  and  part  derived  from  steam  or  from  the  atmosphere,  carrying  with  it 
hydrogen  or  nitrogen.  Multiplying  the  weight  of  each  constituent  gas  in  a  pound 
by  its  calorific  value,  we  have  the  heating  value  of  the  gas.  As  a  mean  of  54 
analyses,  Fernald  finds  (11)  the  following  percentages  by  volume  : 

Carbon  monoxide  (CO) 19.2 

Carbon  dioxide  (CO2) 9.5 

Hydrogen  (H) 12.4 

Marsh  gas  and  ethylene  (CH4,  C2H4) 3.1 

Nitrogen  (H)   . 55.8 

100.0 

286.  Figure  of  Merit.  A  direct  and  accurate  determination  of  efficiency  is 
generally  impossible,  on  account  of  the  difficulties  in  gas  measurement  (12).  For 
comparison  of  results  obtained  from  the  same  coals,  the  figure  of  merit  is  sometimes 
used.  This  is  the  quotient  of  the  heating  value  per  pound  of  the  gas  by  the 
weight  of  carbon  in  a  pound  of  gas :  it  is  the  heating  value  of  the  gas  per  pound  of 
carbon  contained.  In  the  ideal  case,  for  pure  carbon,  its  value  would  be  10,050  B.  t.  u. 
For  a  hydrocarbonaceous  coal,  it  may  have  a  greater  value. 


GAS  ENGINE  CYCLES 

287.  Four-cycle  Engine.  A  gas  engine  of  one  of  the  most  commonly  used 
types  is  shown  in  Fig.  116.  This  represents  a  single-acting  engine;  i.e.  the  gas  is 
in  contact  with  one  side  of  the  piston  only,  the  other  end  being  open.  Large  en- 
gines of  this  type  are  frequently  made  double-acting,  the  gas  being  then  con- 
tained on  both  sides  of  a  piston  moving  in  an  entirely  closed  cylinder,  exhaust 
occurring  on  one  side  while  some  other  phase  of  the  cycle  is  described  on  the 
other  side. 


THE  GAS  ENGINE 


155 


FIG.  116.    Art.  287.  —  Single-acting  Gas  Engine,  Four  Cycle. 
(From  "  The  Gas  Engine,"  by  Cecil  P.  Poole,  with  the  permission  of  the  Hill  Publishing  Company.) 


FIG.  117.     Art.  288.— Piston  Movements,  Otto  Cycle. 
(From  «'The  Gas  Engine,"  by  Cecil  P.  Poole,.  with  the  permission  of  the  Hill  Publishing  Company.) 


156  APPLIED  THERMODYNAMICS 

288.  The  Otto  Cycle.  Figure  117  illustrates  the  piston  move- 
ments corresponding  to  the  ideal  pv  diagram  of  Fig.  118.  The 
cycle  includes  five  distinctly  marked  paths.  During  the  out  stroke 
of  the  piston  from  position  A  to  position  j5,  Fig.  117,  gas  is  sucked 

in  by  its  movement,  giving  the  line 
ai,  Fig.  118.  During  the  next  in- 
ward stroke,  B  to  (7,  the  gas  is  com- 
pressed, the  valves  being  closed, 
along  the  line  be.  The  cycle  is  not 
yet  completed  :  two  more  strokes 
are  necessary.  At  the  beginning 


FIG.  118.    Arts.  288,  291.-TheC      .Cycle.       of    ^    fiwt    of 

being  at  <?,  Fig.  118,  the  gas  is  ignited  and  practically  instantaneous 
combustion  occurs  at  constant  volume,  giving  the  line  cO.  An  out 
stroke  is  produced,  and  as  the  valves  remain  closed,  the  gas  expands, 
doing  work  along  Cd,  while  the  piston  moves  from  0  to  D,  Fig.  117. 
At  d,  the  exhaust  valve  opens,  and  during  the  fourth  stroke  the 
piston  moves  in  from  D  to  E,  expelling  the  gas  from  the  cylinder 
along  c?e,  Fig.  118.  This  completes  the  cycle.  The  inlet  valve  has 
been  open  from  a  to  5,  the  exhaust  valve  from  d  to  e.  During  the 
remainder  of  the  stroke,  the  cylinder  was  closed.  Of  the  four 
strokes,  only  one  was  a  "  working  "  stroke,  in  which  a  useful  effort 
was  made  upon  the  piston.  In  a  double-acting  engine  of  this  type, 
there  would  be  two  working  strokes  in  every  four. 

289.  Two-stroke  Cycle.  Another  largely  used  type  of  engine  is  shown 
in  Fig.  119.  The  same  five  paths  compose  the  cycle ;  but  the  events  are 
now  crowded  into  two  strokes.  The  exhaust  opening  is  at  E ;  no  valve 
is  necessary.  The  inlet  valve  is  at  A,  and  ports  are  provided  at  C,  C  and 
/.  The  gas  is  often  delivered  to  the  engine  by  a  separate  pump,  at  a 
pressure  several  pounds  above  that  of  the  atmosphere ;  in  this  engine,  the 
otherwise  idle  side  of  a  single-acting  piston  becomes  itself  a  pump,  as 
will  appear.  Starting  in  the  position  shown,  let  the  piston  move  to  the  left. 
It  draws  a  supply  of  combustible  gas  through  A,  B  and  the  ports  C  into 
the  chamber  D.  On  the  outward  return  stroke,  the  valve  A  closes,  and  the 
gas  in  D  is  compressed.  Compression  continues  until  the  edge  of  the  piston 
passes  the  port  /,  when  this  high  pressure  gas  rushes  into  the  space  F,  at 
practically  constant  pressure.  The  piston  now  repeats  its  first  stroke. 
Following  the  mass  of  gas  which  we  have  been  considering,  we  find  that 


THE  TWO-CYCLE  ENGINE 


157 


it  undergoes  compression,  beginning  as  soon  as  the  piston  closes  the  ports 
E  and  J,  and  continuing  to  the  end  of  the  stroke,  when  the  piston  is  in  its 
extreme  left-hand  position.  Ignition  there  takes  place,  and  the  next  out 


FIG.  119.    Arts.  289-291,  309,  339.— Two-cycle  Gas  Engine. 
(From  "The  Gas  Engine,"  by  Cecil  P.  Poole,  with  the  permission  of  the  Hill  Publishing  Company.) 

stroke  is  a  working  stroke,  during  which  the  heated  gas  expands.  Toward 
the  end  of  this  stroke,  the  exhaust  port  E  is  uncovered,  and  the  gas  passes 
out,  and  continues  to  pass  out  until  early  on  the  next  backward  stroke  this 
port  is  again  covered. 

290.  Discussion  of  the  Cycle.  We  have  here  a  two-stroke  cyde ;  for 
two  of  the  four  events  requiring  a  perceptible  time  interval  are  always 
taking  place  simultaneously.  On  the  first  stroke  to  the  left,  while  gas  is 
entering  D,  it  is  for  a  brief  interval  of  time  also  flowing  from  /to  F,  from 
F  through  E,  and  afterward  being  compressed  in  F.  On  the  next  stroke 
to  the  right,  while  gas  is  compressed  in  Z>,  ignition  and  expansion  occur  in 
F-,  and  toward  the  end  of  the  stroke,  the  exhaust  of  the  burned  gases 
through  E  and  the  admission  of  a  fresh  supply  through  7,  both  begin. 
The  inlet  port  7  and  the  exhaust  port  E  are  both  open  at  once  during  part 
of  the  operation.  To  prevent,  as  far  as  possible,  the  fresh  gas  from 
escaping  directly  to  the  exhaust,  the  baffle  G  is  fixed  on  the  piston.  It  is 
only  by  skillful  proportioning  of  port  areas,  piston  speed,  and  pressure  in 
D  that  large  loss  from  this  cause  is  avoided.  The  burned  gases  in  the 
cylinder,  it  is  sometimes  claimed,  form  a  barrier  between  the  fresh  enter- 
ing gas  and  the  exhaust  port. 


158 


APPLIED  THERMODYNAMICS 


291.    PV  Diagram.     This  is  shown  for  the  working  side  (space  F)  in 
Fig.  120  and  for  the  pumping  side  (space  D)  in  Fig.  121.     The  exhaust 

port  is  uncovered  at  d,  Fig.  120,  and  the  pres- 
sure rapidly  falls.  At  a,  the  inlet  port  opens, 
the  fresh  supply  of  gas  holding  up  the  pres- 
sure. From  a  out  to  the  end  of  the  diagram, 
and  back  to  b,  both  ports  are  open.  At  b  the 
inlet  port  closes,  and  at  c  the  exhaust  port, 
when  compres-  0  t 

sion  begins.  The 
pump  diagram  of 
Fig.  121  corre- 
sponds with  the 
negative  loop  deab  of  Fig.  118.  Aside  from 


FIG.  120.  Art.  291.— Two-stroke 
Cycle. 


FIG.  121.    Art.  291.—  Two-stroke 
c?cle  PumP  Diagram. 


the  slight  difference  at  dabc,  Fig.  120,  the 

diagrams  for  the  two-cycle  and  four-cycle  engines  are  precisely  the  same  ; 

and  in  actual  indicator  cards,  the  difference  is  very  slight. 


292.  Ideal  Diagram.  The  perfect  PV 
diagram  for  either  engine  would  be  that  of 
Fig.  122,  ebfd,  in  which  expansion  and  com- 
pression are  adiabatic,  combustion  instan- 
taneous, and  exhaust  and  suction  unre- 
strictec^  5  so  that  tne  area  of  the  negative 
loop  dg  becomes  zero,  and  el  and  fd  are 

v  x  i  uii          •  ,• 

lines  of  constant  volume,     r  rom  inspection 
of  the  diagram  we  find 


7,  r/=  v* 

(jr\  v 

y±\  F:  -  v 

Tr       >  *  b—    *  & 


FIG.  122.    Arts.  292,  293,  294, 
295,  314,  329,  331,  Prob.  is.— 

Idealized    Gas   Engine   Dia- 

m 


fTf     /Tf     -L     -f 

"P,' 

293.    Work  Done.     The  work  area  under  6/is  -P* P»  ~ -*/*/.  that 

J>   |7"  j>    TT 

under  gc?  is  — e—^ -^— ^ ;  the  net  work  of  the  cycle  is 

y-l  * 


EFFICIENCY   OF   OTTO   CYCLE 


159 


This  may  be  written  in  terms  of  two  pressures  and  two  volumes  only, 
for  P,  Ve  =  Pd  Vj>  V?-*  and  PfVf  =  Pb  V/  VJ-»,  giving 


y-J 


P      fV  V          P      fV\y 

294.  Relations  of  Curves.     Expressing  -^  =  f  -^  j   and  —  =  f  -^  J ,  and 

P      P  P      P 

remembering  that  F6  =  Ve,  Vf=  Vd,  we  have  -^  ==  -=?-  and  ^  =  ^ .     This 

permits  of  rapidly  plotting  one  of  the  curves  when  the  other  is  given. 
We  also  find  ^r  =  ^f  and  ^r=jr- 

295.  Efficiency.     In  Fig.  122,  heat  is  absorbed  along  eb,  equal  to 
l(Tb  —  Te);  this  is  derived  from  the  combustion  of  the  gas.     Heat 
is  rejected  along  fd,  =l(Tf—  TJ).     Using  the  difference  of  the  two 
quantities  as  an  expression  for  the  work  done,  we  obtain  for  the 
efficiency 

FTI         fjj  FJJ 

Tb-Te=     ~  Tb 


Tb-Te 


=  . 


T 


Te 


The  efficiency  thus  depends  solely  upon  the  extent  of  compression. 

296.  Carnot  Cycle  and  Otto  Cycle;  the  Atkinson  Engine.  Let  abed, 
Fig.  123,  represent  a  Carnot  cycle  drawn  to  pv  coordinates,  and  bfde,  the 
corresponding  Otto  cycle  between  the 
same  temperature  limits,  T7  and  t.  For  the 
Carnot  cycle,  the  efficiency  is  (T  —  t)  -i-  T; 
for  the  Otto,  it  is,  as  has  been  shown, 
(Te—  Td)  -=-  Te.  It  is  one  of  the  disad- 
vantages of  the  Otto  cycle,  as  shown  in 
Art.  294,  that  the  range  of  temperatures 
during  expansion  is  the  same  as  that  dur- 
ing compression.  In  the  ingenious  Atkin- 
son engine  (13),  the  fluid  was  contained  in 
the  space  between  two  pistons,  which  space 
was  varied  during  the  phases  of  the  cycle.  This  permitted  of  expansion 
independent  of  compression  ;  in  the  ideal  case,  expansion  continued  down 


FIG.  123.    Art.  296.— Carnot,  Otto, 
and  Atkinson  Cycles. 


160 


APPLIED  THERMODYNAMICS 


to  the  temperature  of  the  atmosphere,  giving  such  a  diagram  as  ebcd,  Fig. 

123.  The  entropy  diagrams  for  the  Carnot,  Otto,  and  Atkinson  cycles  are 

correspondingly  lettered  in  Fig.  124.  For 
the  Atkinson  cycle,  in  the  ideal  case,  we 
have  in  Fig.  124  the  elementary  strip 
vwxy,  which  may  stand  for  dH,  and  the 
isothermal  dc  at  the  temperature  t.  Let 
the  variable  temperature  along  eb  be  Tx, 
having  for  its  limits  Tb  and  Te.  Then  for 
the  area  ebcd,  we  have 


FIG.  124.     Arts.  290,  297,  305,  307.— 
Efficiencies  of  Gas  Engine  Cycles. 


The  efficiency  is  obtained  by  dividing  by  I  (Tb  —  Te)  and  is  equal  to 

_     _J lo     T> 

297.    Application  to  a  Special  Case.     Let    Te  =  1060,    T  =  3440,     t  =  520 ; 
whence,  from  Art.  294,  T/  =  1688.     We  then  have  the  following  ideal  efficiencies: 

T  -  t      3440  -  520 


Carnot, 


Atkinson,  1  — 


Otto, 


3440 
520 


=  0.85. 


Tb- 


Te  -  t      1060  -  520 


=  0.51. 


Te  1060 

The  Atkinson  engine  can  scarcely  be  regarded  as  a  practicable  type ;  the 
Otto  cycle  is  that  upon  which  most  gas  engine  efficiencies  must  be  based ; 
and  they  depend  solely  on  the  ratio  of  temperatures  or  pressures  during 
compression. 

298.  Lenoir  Cycle.  This  is  shown  in  Fig.  125.  The  fluid  is  drawn 
into  the  cylinder  along  Ad  and  exploded  along  df.  Expansion  then 
occurs,  giving  the  pathjft/,  when  the  exhaust  valve  opens,  the  pressure 


STANT  VOLUME 


FlG.  125.     Arts.  298,  301,  302.— 
Lenoir  Cycle. 


FIG.  126.     Art.  298. —Entropy 
Diagram,  Lenoir  Cycle. 


BRAYTON  CYCLE 


161 


falls,  gh,  until  it  reaches  that  of  the  atmosphere,  and  the  gases  are  finally 
expelled  on  the  return  stroke,  hA.     It  is  a  two-cycle,  engine.     The   net 
entropy  diagram  appears  in  Fig.  126. 
The  efficiency  is 

Heat  absorbed  -  heat  rejected  _l(Tf-  Td}  -  l(Tg  -  Th)  -  k(Th  -  Tg) 
Heat  absorbed  /(7>-  Td) 

^         T.-Tk         Th-Td 
~Tf-Td     yTf-Td 

299.  Bray  ton  Cycle.  This  is  shown  in  Fig.  127.  A  separate 
pump  is  employed.  The  substance  is  drawn  in  along  Ad,  compressed 
along  dn,  and  forced  into  a  reservoir  along  nB.  The  engine  begins 
to  take  a  charge  from  the  reservoir  at  B,  which  is  slowly  fed  in  and 
ignited  as  it  enters,  so  that  combustion  proceeds  at  the  same  rate  as 
the  piston  movement,  giving  the  constant  pressure  line  Bb.  Expan- 


FIG.   127.      Arts.   299,  302.— Bray  ton 
Cycle. 


FIG.  128.    Art.  299.  —  Bray  ton  Cycle, 
Entropy  Diagram. 


sion  then  occurs  along  hg,  the  exhaust  valve  opens  at  g,  and  the 
charge  is  expelled  along  hA.  The  net  cycle  is  dnbgh;  the  net  ideal 
entropy  diagram  is  as  in  Fig.  128.  This  is  also  a  two-cycle 
engine.  The  "  constant  pressure  "  cycle  which  it  uses  was  suggested 
in  1865  by  Wilcox.  In  1873,  when  first  introduced  in  the  United 
States,  it  developed  an  efficiency  of  2.7  Ib.  of  (petroleum)  oil  per 
brake  .hp.-hr. 

The  efficiency  is  (Fig.  127) 


If  expansion  is  complete,  the  cycle  becoming  dribi,  Figs.  127,  128,  then 
Tg  =  Th  =  Ti,  and  the  efficiency  is 


1- 


-  T 


Tn-  Td 


-  Tn  Tn          Tn 

a  result  identical  with  that  in  Art.  295 ;  the  efficiency  (with  complete  ex- 
pansion) depends  solely  upon  the  extent  of  compression. 


162  APPLIED  THERMODYNAMICS 

300.  Comparisons  with  the  Otto  Cycle.    It  is  proposed  to  compare  the  capacities 
and  efficiencies  of  engines  working  in  the  Otto,*  Brayton,  and  Lenoir  cycles;  the 
engines  being  of  the  same  size,  and  working  between  the  same  limits  of  temperature. 
For  convenience,  pure  air  will  be  regarded  as  the  working  substance.     In  each  case 
let  the  stroke  be  2  ft.,  the  piston  area  1  sq.  ft.,  the  external  atmosphere  at  17°  C., 
the  maximum  temperature  attained,  1537°  C.     In  the  Lenoir  engine,  let  ignition 
occur  at  half  stroke;  in  the  Brayton,  let  compression  begin  at  half  stroke  and  con- 
tinue until  the  pressure  is  the  same  as  the  maximum  pressure  attained  in  the  Lenoir 
cycle,  and  let  expansion  also  begin  at  half  stroke.     These  are  to  be  compared  with 
an  Otto  engine,  in  which  the  pump  compresses  1  cu.  ft.  of  free  air  to  40  lb.  net 
pressure.     This  quantity  of  free  air,  1  cu.  ft.,  is  then  supplied  to  each  of  the  three 
engines. 

301.  Lenoir  Engine.     The  expenditure  of  heat  (in  work  units)  along  df,  Fig. 
125,  is  Jl(T -  t),  in  which  T  =  1537,  t  =  17,J  is  the  mechanical  equivalent  of  a 
Centigrade    heat   unit,   and  I   is    the    specific    heat    of    1   cu.  ft.  of    free   air, 
heated  at  constant  volume  1°  C.     Now  /  =  778  x  f  =  1400.4,  and  Jl  is  approxi- 
mately 0.1689  x  0.075  x  1400.4  =  17.72.     The  expenditure  of  heat  is  then 

17.72(1537  -  17)  =  26,900  ft.-lb. 
The  pressure  at /is 

-,,  71537  +  273      01  .  ,,      ,     ,   , 
14.7  — -t — —  =  91.4  lb.  absolute; 

and  the  pressure  at  g  is 

91.4(|)y  =  34.25  lb.  absolute. 
The  work  done  under  fg  is  then 

f  /'Ql  4.  v   "M       C\£,  ')£»  v  9^  1 

-I  A  A          1*7 1.*    A     .11  —  I  O-±. £*J    X    £)  Q1  ftA    f4-      IV 

1«  i  •* * » —  \  =  oI90  It. -ID. 

The  negative  work  under  Tid  is  14.7  x  144  x  1  =  2107  ft.-lb.,  and  the  net  work  is 
8190  -  2107  =  6083  ft.-lb.     The  efficiency  is  then  6083  -*•  26,900  =  0.22Q. 

302.  Brayton  Engine.     We  first  find  (Fig.  127) 

Tn  =Td(^\  y   =  (273  -f  17) f ^4V'286  =  489°  absolute  or  216°  C. 


Proceeding  in  the  same  way  as  with  the  Lenoir  engine,  we  find  the  heat  expendi- 
ture to  be 

Jk(Tb  -  ?;)=  0.2375  x  0.075  x  1400.4(1537  -  216)-  33,000  ft.-lb. 

The  pressure  at  n  is  by  assumption  equal  to  that  in  the  case  of  the  Lenoir  engine ; 
the  pressure  at  g  in  the  Brayton  type  then  equals  that  at  g  in  the  Lenoir.  The 
work  under  bg  is  the  same  as  that  under  fg  in  Fig.  125.  The  work  under  nb  is 
found  by  first  ascertaining  the  volume  at  n.  This  is 


f—  V:V1.0  =0.272. 
U.14/ 


*  The  "  Otto  cycle  "  in  this  discussion  is  a  modified  form  (as  suggested  by  Clerk) 
in  which  the  strokes  are  of  unequal  length. 


CLERK'S  GAS   ENGINE  163 

The  work  under  nb  is  then  91.4  x  144  x  (1.0  -  0.272)  =  9650  ft.-lb.,  and  the  gross 
work  is  9650  +  8190  -  17,840  ft.-lb.  Deducting  the  negative  work  under  hd, 
2107  ft.-lb.,  and  that  under  dn, 

-  -     144 


the  net  work  area  is  12,083  ft.-lb.,  and  the  efficiency,  12,083  -f-  33,000  =  0.366. 

303.  Clerk's  Otto  Engine.  In  Fig.  129,  a  separate  pump  takes  in  a  charge 
along  AB,  and  compresses  it  along  BC,  afterward  forcing  it  into  a  receiver  along 
CD  at  40  Ib.  gauge  pressure.  Gas  flows  from 
the  receiver  into  the  engine  along  DC,  is  ex- 
ploded along  CE,  expands  to  F,  and  is  expelled 
along  GA.  The  net  cycle  is  BCEFG.  The  D 
volume  at  C  is 

(if^ Y"y=  0.393  cu.  ft. 

FIG.  129.     Arts.  303,  305.  — Clerk's 

The  temperature  at  C  is  otto  °ycle- 

PVT^PV-  (54-7  x  0-393)  (273  +  17)  _  Q73  _  153o  n 

14.7  x  1 
The  pressure  at  E  is  then 

(1537  +  273)54.7 

— ^ =231  Ib.  absolute. 

Io3  +  273 

The  pressure  at  F  is 

231  ( M^V  =  23.64  Ib.  absolute. 


The  work  under  EF  is 

144  /  (231  x  0.393)-  (23.64  x  2)  \  _  -  f     , 

\  1.402  -  1.0  )  ft~lb'' 

that  under  BG  is  2107  ft.-lb.,  and  that  under  BC  is 

\  =  243Q  ft>_l 


54.7  x  0.898)^(14.  7  xl)\ 


The  net  work  is  15,600  -  2107  -  2430  =  11,063  ft.-lb.  The  heat  expenditure  in 
this  case  is  Jl(TE  -  Tc}  =  17.72  x  (1537  -  153)  =  24,500  ft.-lb.,  and  the  efficiency 
is  11,063  H-  24,500  =  0.453;  considerably  greater  than  that  of  either  the  Lenoir  or 
the  Brayton  engine  (14).  If  we  express  the  cyclic  area  as  100,  then  that  of  the 
Lenoir  engine  is  52  and  that  of  the  Brayton  engine  is  104. 

304.  Trial  Results.  These  comparisons  correspond  with  the  consumption  of 
gas  found  in  actual  practice  with  the  three  types  of  engine.  The  three  efficiencies 
are  0.226,  0.366,  and  0.453.  Taking  4  cu.  ft.  of  free  gas  as  ideally  capable  of  giv- 
ing one  horse  power  per  hour,  the  gas  consumption  per  hp.-hr.  in  the  three  cases 
would  be  respectively  4  -s-  0.226  =  17.7,  4  -f-  0.366  =  10.9,  and  4  -f-  0.453  =  8.84  cu.  ft, 
Actual  tests  gave  for  the  Lenoir  and  Hugon  engines  90  cu.  ft.  ;  for  the  Brayton, 
50  ;  and  for  the  modified  Otto,  21.  The  possibility  of  a  great  increase  in  economy 


164  APPLIED  THERMODYNAMICS 

by  the  use  of  an  engine  of  a  form  somewhat  similar  to  that  of  the  Brayton  will  be 
discussed  later. 

305.    Complete  Pressure  Cycle.     The  cycle  of  Art.  303  merits  detailed  exami- 
nation.    In  Fig.  129,  the  heat  absorbed  is  l(TE  —  TC)  ;  that  rejected  is 


the  efficiency  is 

"  TE  -  Tc 

The  entropy  diagram  may  be  drawn  as  ebmnd,  Fig.  124,  showing  this  cycle  to  be 
more  efficient  than  the  equal-length-stroke  Otto  cycle,  but  less  efficient  than  the 
Atkinson.  With  complete  expansion  down  to  the  lower  pressure  limit,  the  cycle 
becomes  BCEFH,  Fig.  129,  or  ebort,  Fig.  124;  the  strokes  are  still  of  unequal 
length,  and  the  efficiency  is  (Fig.  129) 

TH-TR 

TE  -  Tc 

If  the  strokes  be  made  of  equal  length,  with  incomplete  expansion,  TG=  T£,  the 
cycle  becomes  the  ordinary  Otto,  and  the  efficiency  is 

..       7V  —  Ta  _  TC  —  T  n 


306.  Oil  Engines :  The  Diesel  Cycle.  Oil  engines  may  operate  in  either 
the  two-stroke  or  the  four-stroke  cycle,  usually  the  latter ;  and  combus- 
tion may  occur  at  constant  volume  (Otto),  constant  pressure  (Brayton),  or 
constant  temperature  (Diesel).  Diesel,  in  1893  (15),  first  proposed  what 
has  proved  to  be  from  a  thermal  standpoint  the  most  economical  heat 
engine.  It  is  a  four-cycle  engine,  approaching  more  closely  than  the 
Otto  to  the  Carnot  cycle,  and  theoretically  applicable  to  solid,  liquid,  or 

gaseous  fuels,  although  actually  used  only 
with  oil.  The  first  engine,  tested  by  Schroter 
in  1897,  gave  indicated  thermal  efficiencies 
ranging  from  0.34  to  0.39  (16).  The  ideal- 
ized cycle  is  shown  in  Fig.  130.  The  opera- 
tions are  adiabatic  compression,  isothermal 

expansion,    adiabatic    expansion,    and    dis- 
FIG.  130.     Arts.  306,  307.  — Diesel          *  .  '     .     . 

Cycle  charge  at  constant  volume.     Jrure  air  is  com- 

pressed to  a  high  pressure  and  temperature, 

and  a  spray  of  oil  is  then  gradually  injected  by  means  of  external  air 
pressure.  The  temperature  of  the  cylinder  is  so  high  as  at  once  to  ignite 
the  oil,  the  supply  of  which  is  so  adjusted  as  to  produce  combustion 
practically  at  constant  temperature.  Adiabatic  expansion  occurs  after 
the  supply  of  fuel  is  discontinued.  A  considerable  excess  of  air  is  used. 
The  pressure  along  the  combustion  line  is  from  30  to  40  atmospheres,  that 


THE  DIESEL  ENGINE 


165 


at  which  the  oil  is  delivered  is  50  atmospheres,  and  the  temperature  at 
the  end  of  compression  approaches  1000°  F.  The  engine  is  started  by 
compressed  air ;  two  or  more  cylinders  are  used.  There  is  no  uncertainty 
as  to  the  time  of  ignition ;  it  begins  immediately  upon  the  entrance  of 


FIG.  131.     Art.  306.  —  Diesel  Engine.     (American  Diesel  Engine  Company.) 

the  oil  into  the  cylinder.  To  avoid  pre-ignition  in  the  supply  tank,  the 
high-pressure  air  used  to  inject  the  oil  must  be  cooled.  The  cylinder 
is  water-jacketed.  Figure  131  shows  a  three-cylinder  engine  of  this  type ; 
Fig.  132,  its  actual  indicator  diagram,  reversed. 


166 


APPLIED  THERMODYNAMICS 


FIG.  132.      Art.  30(>.  —  Indicator  Diagram,  Diesel  Engine. 
(16  x  24  in.  engine,  160  r.p.m.     Spring  400.) 

307.    Efficiency.     The  heat  absorbed  along  ab,  Fig.  130,  is 
p  Y  los-  ^=RT  loo-  -Zk 

-L  a  '  a  iua«    TT-  —  •*«'•*«  Jut>e    y 

'a  *  a, 

The  heat  rejected  along/c?  is  l(Tj—  T^.     We  may  write  the  efficiency 
as 


But  2>t=  Tb(^}      =  Ta(j?)      >and  Ta=Tdl-^}      ;  whence 

rp  rp  I       f\       I       b\  —  HI  (       b\ 

•Lf~       d\V  }         VT^y  ~       d\  V  I 

\vfj     \py  \vj 

For  the  heat  rejected  along  fd  we  may  therefore  write 

—  f    (    h]      1 

and  for  the  efficiency, 

1- 


Muf-y.r  - 


V 

This  increases  as  3T0  increases  and  as  — -  decreases.     The  last  conclu- 

* a 

sion  is  of  prime  importance,  indicating  that  the  efficiency  should  in- 
crease at  light  loads.  This  may  be  apprehended  from  the  entropy 
diagram,  abfd,  Fig.  124.  As  the  width  of  the  cycle  decreases  (If 
moving  toward  ad),  the  efficiency  increases. 


MODIFICATIONS  OF  THE  OTTO   CYCLE 


167 


In  constructing  the  entropy  diagram  from  an  actual  Diesel  indicator  card 
a  difficulty  arises  similar  to  one  met  with  in  steam  engine  cards;  the  quantity  of 
substance  in  the  cylinder  is  not  constant  (Art.  454).  This  has  been  discussed 
by  Eddy  (17),  Frith  (18),  and  Reeve  (19). 
The  illustrative  diagram,  constructed  as 
in  Art.  317,  is  suggestive.  Figure  133 
shows  such  a  diagram,  for  an  engine 
tested  by  Denton  (20).  The  initially  hot 
cylinder  causes  a  rapid  absorption  of  heat 
from  the  walls  during  the  early  part  of 
compression  along  ab.  Later,  along  be, 
heat  is  transferred  in  the  opposite  direc- 
tion. Combustion  occurs  along  cd,  the 
temperature  and  quantity  of  heat  increas- 
ing rapidly.  During  expansion,  along  de, 
the  temperature  falls  with  increasing 
rapidity,  the  path  becoming  practically  adiabatic  during  release,  along  ef.  The 
TV  diagram  of  Fig.  133  indicates  that  no  further  rise  of  temperature  would  ac- 
company increased  compression ;  the  actual  path  at  y  has  already  become  prac- 
tically isothermal. 

308.  Comparison  of  Cycles.  Figure  134  shows  all  of  the  cycles  that  have 
been  discussed,  on  a  single  pair  of  diagrams.  The  lettering  corresponds 
with  that  in  Figs.  122-128,  130.  The  cycles  are, 

Carnot,  abed,  Lenoir,  df^Ji^df^  Diesel,  dabf, 

Otto,  ebfd,  Brayton,  dnbgh,  dnbi,          Atkinson,  ebcd, 

Complete  pressure,  debgh,  debt. 


FIG,  133. 


Art.  307.  —Diesel  Engine 
Diagrams. 


FIG.  134.    Art.  308,  Probs.  7,  25.  — Comparison  of  Gas  Engine  Cycles. 


PRACTICAL  MODIFICATIONS  OF  THE  OTTO  CYCLE 

309.  Importance  of  Proper  Mixture.  The  working  substance  used  in  gas 
engines  is  a  mixture  of  gas,  oil  vapor  or  oil,  and  air.  Such  mixtures  will  not 
ignite  if  too  weak  or  too  strong.  Even  when  so  proportioned  as  to  permit  of 
ignition,  any  variation  from  the  ideally  correct  ratio  has  a  detrimental  effect;  if 


168  APPLIED  THERMODYNAMICS 

too  little  air  is  present,  the  gas  will  not  burn  completely,  the  exhaust  will  be  dark- 
colored  and  odorous,  and  unburned  gas  may  explode  in  the  exhaust  pipe  when 

it  meets  more  air.  If  too  much  air  is  admitted, 
the  products  of  combustion  will  be  unnecessarily 
diluted  and  the  rise  of  temperature  during 
ignition  will  be  decreased,  causing  a  loss  of  work 
area  on  the  PV  diagram.  Figure  135  shows  the 
effect  on  rise  of  temperature  and  pressure  of 
varying  the  proportions  of  air  and  gas,  assuming 
the  variations  to  remain  within  the  limits  of 

" 


!to.  185.     Art.  309.-  Effect  ol 

Mixture  Strength.  as   a  result  of  tne  presence  of  excess  of  air   as 

well  as  when  the  air  supply  is  deficient.    Rapidity 

of  flame  propagation  is  essential  for  efficiency,  and  this  is  only  possible  with  a 
proper  mixture.  The  gas  may  in  some  cases  burn  so  slowly  as  to  leave  the  cyl- 
inder partially  unconsumed.  In  an  engine  of  the  type  shown  in  Fig.  119,  this 
may  result  in  a  spread  of  flame  through  /,  B,  and  C  back  to  D,  with  dangerous 
consequences. 

310.  Methods  of  Mixing.     The  constituents  of  the  mixture  must  be  intimately 
mingled  in  a  finely  divided  state,  and  the  governing  of  the  engine  should  prefer- 
ably be  accomplished  by  a  method  which  keeps  the  proportions  at  those  of  highest 
efficiency.     Variations  of  pressure  in  gas  supply  mains  may  interpose  serious  dif- 
ficulty in  this  respect.     Fluctuations  in  the  lights  which  may  be  supplied  from  the 
same  mains  are  also  excessive  as  the  engine  load  changes.     Both  difficulties  are 
sometimes  obviated  in  small  units  by  the  use  of  a  rubber  supply  receiver.     Varia- 
tions in  the  speed  of  the  engine  often  change  the  proportions  of  the  mixture. 
When  the  air  is  drawn  from  out  of  doors,  as  with  automobile  engines,  variations 
in  the  temperature  of  the  air  affect  the  mixture  composition.     In  simple  types  of 
engine,  the  relative  openings  of  the  automatic  gas  and  air  inlet  valves  are  fixed 
when  the  engine  is  installed,  and  are  not  changed  unless  the  quality  or  pressure 
of  the  gas  changes,  when  a  new  adjustment  is  made  by  the  aid  of  the  indicator  or 
by  observation  of  the  exhaust.     Mechanically  operated  valves  are  used  on  high- 
speed engines;  these  are  positive  in  their  action.     The  use  of  separate  pumps  for 
supplying  air  and  gas  permits  of  proportioning  in  the  ratio  of  the  pump  displace- 
ments, the  volume  delivered  being  constant,  regardless  of  the  pressure  or  tempera- 
ture.    Many  adjustable   mixing  valves  and  carburetors  are  made,  in  which  the 
mixture  strength  may  be  regulated  at  will.     These  are  necessary  where  irregulari- 
ties of  pressure  or  temperature  occur,  but  require  close  attention  for  economical 
results.     The  presence  of  burned  gas  in  the  clearance  space  of  the  cylinder  affects 
the  mixture,  retarding  the  flame  propagation.     The  effect  of  mixture  strength  on 
allowable  compression  pressures  remains  to  be  considered. 

311.  Actual  Gas  Engine  Diagram.     A  typical  indicator  diagram  from 
a  good  Otto  cycle  engine  is  shown  in  Fig.  136.     The  various  lines  differ 
somewhat  from  those  established  in  Art.  288.     These  differences  we  now 
discuss.     Figure  137  shows  the  portion  bcde  of  the  diagram  in  Fig.  136 
to  an  enlarged  vertical  scale,  thus  representing  the  action  more  clearly. 


ALLOWABLE  COMPRESSION  169 

The  line  fg  is  that  of  atmospheric  pressure,  omitted  in  Fig.  136.     We  will 
begin  our  study  of  the  actual  cycle  with  the  compression  line. 


FIG.  136.     Arts.  311,  342,  £45.  —  FIG.  137.     Arts.   311,   326,   328.  —  Eu- 

Otto  Engine  Indicator  Diagram.  larged  Portion  of  Indicator  Diagram. 

312.  Limitations  of  Compression.  It  has  been  shown  that  a  high  degree 
of  compression  is  theoretically  essential  to  economy.  In  practice,  com- 
pression must  be  limited  to  pressures  (and  corresponding  temperatures) 
at  which  the  gases  will  not  ignite  of  themselves ;  else  combustion  will 
occur  before  the  piston  reaches  the  end  of  the  stroke,  and  a  backward 
impulse  will  be  given.  Gases  differ  widely  as  to  the  temperatures  at 
which  they  will  ignite ;  hydrogen,  fft  example,  inflames  so  readily  that 
Lucke  (21)  estimates  that  the  allowable  final  pressure  must  be  reduced 
one  atmosphere  for  each  5  per  cent  of  hydrogen  present  in  a  mixed  gas. 

The  following  are  the  average  final  gauge  compression  pressures 
recommended  by  Lucke  (22)  :  for  gasoline,  in  automobile  engines, 
45  to  95  lb.,  in  ordinary  engines,  60  to  85  Ib.  ;  for  kerosene,  30  to  85  lb.; 
for  natural  gas,  75  to  130  lb.  ;  for  coal  gas  or  carbureted  water  gas, 
60  to  100  lb.  ;  for  producer  gas,  100  to  160  lb.  ;  and  for  blast  furnace 
gas,  120  to  190  lb.  The  range  of  compression  depends  also  upon  the 
pressure  existing  in  the  cylinder  at  the  beginning  of  compression ;  for 
two-cycle  engines,  this  varies  from  18  to  21  lb.,  and  for  four-cycle 
engines,  from  12  to  14  lb.,  both  absolute. 

The  pre-compression  temperature  also  limits  the  allowable  range  below  the 
point  of  self-ignition.  This  temperature  is  not  that  of  the  entering  gases,  but  it 
is  that  of  the  cylinder  contents  at  the  moment  when  compression  begins ;  it  is 
determined  by  the  amount  of  heat  given  to  the  incoming  gases  by  the  hot  cylin- 
der walls,  and  this  depends  largely  upon  the  thoroughness  of  the  water  jacketing 
and  the  speed  of  the  engine.  This  accounts  for  the  rather  wide  ranges  of  allow- 
able compression  pressures  above  given.  Usual  pre-compression  temperatures  are 
from  140°  to  300°  F.  "Scavenging"  the  cylinder  with  cold  air,  the  injection  of 
water,  or  the  circulation  of  water  in  tubes  in  the  clearance  space,  may  reduce  this. 
Usual  practice  is  to  thoroughly  jacket  all  exposed  surfaces,  including  pistons 
and  valve  faces,  and  to  avoid  pockets  where  exhaust  gases  may  collect.  The 
primary  object  of  jacketing,  however,  is  to  keep  the  cylinder  cool,  both  for  me- 
chanical reasons  and  to  avoid  uncontrollable  explosions  at  the  moment  when  the 
gas  reaches  the  cylinder. 


170  APPLIED  THERMODYNAMICS 

313.  Practical    Advantages    of    Compression.     Compression    pressures    have 
steadily  increased  since  1881,  and  engine  efficiencies  have  increased  correspond- 
ingly, although  the  latter  gain  has  been  in  part  due  to  other  causes.     Improved 
methods  of  ignition  have  permitted  of  this  increased  compression.     Besides  the 
therrnodynamic  advantage  already  discussed,  compression  increases  the  engine 
capacity.     In  a  non-compressive  engine,  no  considerable  range  of  expansion  could 
be  secured  without  allowing  the  final  pressure  to  fall  too  low  to  give  a  large  work 
area;   in  the  compressive  engine,  wide  expansion  limits  may  be  obtained  along 
with  a  fairly  high  terminal  pressure.     Compression  reduces  the  exposed  cylinder 
surface  in  proportion  to  the  weight  of  gas  present  at  maximum  temperature,  and 
so  decreases  the  loss  of  heat  to  the  walls.     The  decreased  proportion  of  clearance 
space  following  the  use  of  compression  also  reduces  the  proportion  of  spent  gases 
to  be  mixed  with  the  incoming  charge. 

314.  Pressure  Rise  during  Combustion.     In  Art.  292,  .the  pressure  P6  after 
combustion  was  assumed.     While,  for  reasons  which  will  appear,  any  computation 
of  the  rise  of  pressure  by  ordinary  methods  is  unreliable,  the  method  should  be 
described.     Let  H  denote  the  amount  of  heat  liberated  by  combustion,  per  pound 

of  fuel.     Then,  Fig.  122,  H=  l(Tb  -  *),    Tb  -  Te  =  -  and  Tb  =  —  +  Te.     But 


Pe  _k-l 

rTe~7v^ 

i(^]yvd 
\PJ    d 

Then  Pb-P  =g^-0=     a402^ 


315.  Computed  Maximum  Temperature.  Dealing  now  with  the  constant 
volume  ignition  line  of  the  ideal  diagram,  let  the  gas  be  one  pound  of  pure 
carbon  monoxide,  mixed  with  just  the  amount  of  air  necessary  for  com- 
bustion (2.48  lb.),  the  temperature  at  the  end  of  compression  being  1000° 
absolute,  and  the  pressure  200  lb.  absolute.  Since  the  heating  value  of  1 
lb.  of  CO  is  4315  B.  t.  u.,  while  the  specific  heat  at  constant  volume  of 
C02  is  0.1692,  that  of  N  being  0.1727,  we  have 


rise  in  temperature  =  -        -  ———  =  7265°  F. 
(1.57  x  0.1692)+  (1.91  x  0.1727) 

The  temperature   after  complete  ignition  is  then  8265°  absolute.     The 

pressure  is  200  x  —   —  =  1653  lb.      If  the  volume  increases  during  igni- 
lUuu 

tion,  the  pressure  decreases.  Suppose  the  volume  to  be  doubled,  the  rise 
of  temperature  being,  nevertheless,  as  computed  :  then  the  maximum  pres- 
sure attained  is  826.5  lb. 


DISSOCIATION  171 

316.  Actual  Maxima.     No  such  temperature  as  8265°  absolute  is  at- 
tained.   In  actual  practice,  the  temperature  after  ignition  is  usually  about 
3500°  absolute,  and  the  pressure  under  400  Ib.     The  rise  of  either  is  less 
than  half  of  the  rise  theoretically  computed,  for  the  actual  air  supply, 
with  the  actual  gas  delivered.    It  is  difficult  to  measure  the  maximum  tem- 
perature, on  account  of  its  extremely  brief  duration.     It  is  more  usual  to 
measure  the  pressure  and  compute  the  temperature.     This  is  best  done  by 
a  graphical  method,  as  with  the  indicator. 

• 

317.  Explanation  of  Discrepancy.     There  are  several  reasons  for  the  disagree- 
ment between  computed  and  observed  results.     Charles'  law  does  not  hold  rigidly 
at  high  temperatures ;  the  specific  heats  of  gases  are  known  to  increase  with  the 
temperature  (Meyer  found  in  one  case  the  theoretical  maximum  temperature  to 
be  reduced  from  4250°  F.  to  3330°  F.  by  taking  account  of  the  increases  in  specific 
heats  as  determined  by  Mallard  and  Le  Chatelier);   combustion  is  actually  not 
instantaneous  throughout  the  mass  of  gas  and  some  increase  of  volume  always 
occurs  ;  and  the  temperature  is  lowered  by  the  cooling  effect  of  the  cylinder  walls. 
Still  another  reason  for  the  discrepancy  is  suggested  in  Art.  318. 

318.  Dissociation.     Just  as  a  certain  maximum  temperature  must  be  attained 
to  permit  of  combustion,  so  a  certain  maximum  temperature  must  not  be  exceeded 
if  combustion  is  to  continue.     If  this  latter  temperature  is  exceeded,  a  suppression 
of  combustion  ensues.     Mallard  and  Le  Chatelier  found  this  "dissociation  "  effect 
to  begin  at  about  3200°  F.  with  carbon  monoxide  and  at  about  4500°  F.  with  steam. 
Deville,  however,  found  dissociative  effects  with  steam  at  1800°  F.,  and  with  car- 
bon dioxide  at  still  lower  temperatures.     The  effect  of  dissociation  is  to  produce, 
at  each  temperature  within  the  critical  range  for  the  gas  in  question,  a  stable 
ratio  of  combined  to  elementary  gases,  —  e.g.  of  steam  to  oxygen  and  hydrogen, — 
which  cannot  widely  vary.     No  exact  relation  between  specific  temperatures  and 
such  stable  ratio  has  yet  been  determined.     It  has  been  found,  however,  that  the 
maximum  temperature  actually  attained  by  the  combustion  of  hydrogen  in  oxygen 
is  from  3500°  to  3800°  C.,  although  the  theoretical  temperature  is  about  9000°  C. 
At  constant  pressure  (the  preceding  figures  refer  to  combustion  at  constant  vol- 
ume), the  actual  and  theoretical  figures  are  2500°  and  6000°  C.  respectively.     For 
hydrogen  burning  in  air,  the  figures  are  1830  to  2000°,  and  3800°  C.     Dissociation 
here  steps  in  to  limit  the  complete  utilization  of  the  heat  in  the  fuel.     In  gas  en- 
gine practice,  the  temperatures  are  so  low  that  dissociation  cannot  account  for  all 
of  the  discrepancy  between  observed  and  computed  values ;  but  it  probably  plays 
a  part. 

319.  Rate  of  Flame  Propagation.     This  has  been  mentioned  as  a  factor  influ- 
encing the  maximum   temperature  and  pressure  attained.     The  speed  at  which 
flame  travels  in  an  inflammable  mixture,  if  at  rest,  seldom  exceeds  65  ft.  per  sec- 
ond.    If  under  pressure  or  agitation,  pulsations  may  be  produced,  giving  rise  to 
"explosion  waves,"  in  which  the  velocity  is  increased  and  excessive  variations  in 
pressure  occur,  as  combustion  is  more  or  less  localized  (23).     Clerk  (24),  experi- 


172 


APPLIED  THERMODYNAMICS 


meriting  on  mixtures  of  coal  gas  with  air,  found  maximum  pressure  to  be  obtained 
in  minimum  time  when  the  proportion  of  air  to  gas  by  volume  was  5  or  6  to  1 : 
for  pure  hydrogen  and  air,  the  best  mixture  was  5  to  2.  The  Massachusetts  Insti- 
tute of  Technology  experiments,  made  with  carbureted  water  gas,  showed  the  best 
mixture  to  be  5  to  1 ;  with  86°  gasoline,  the  quickest  inflammation  was  obtained 
when  0.0217  parts  of  gasoline  were  mixed  with  1  part  of  air;  with  76°  gasoline, 
when  0.0263  to  0.0278  parts  were  used.*  Grover  found  the  best  mixture  for  coal 
gas  to  be  7  to  1 ;  for  acetylene,  7  or  8  to  1,  acetylene  giving  higher  pressures  than 
coal  gas.  With  coal  gas,  the  weakest  ignitible  mixture  was  15  to  1,  the  theoreti- 
cally perfect  mixture  being  5.7  to  1.  The  limit  of  weakness  with  acetylene  was  18 
to  1.  Both  Grover  and  Lucke  (26)  have  investigated  the  effect  of  the  presence  of 
"  neutrals  "  (carbon  dioxide  and  nitrogen,  derived  either  from  the  air,  the  incom- 
ing gases,  or  from  residual  burnt  gas)  on  the  rapidity  of  propagation.  The  re- 


4.6       5       5.5 
PARTS  AIR  PER  ONE  PART  GAS 


FIG.  138.    Art.  319.  —Effect  of  Presence  of  Neutrals. 
(From  Button's  "The  Gas  Engine,"  by  permission  of  John  Wiley  &  Sons,  Publishers.) 

suits  of  Lucke's  study  of  water  gas  are  shown  in  Fig.  138.  The  ordinates  show 
the  maximum  pressures  obtained  with  various  proportions  of  air  and  gas.  These 
are  highest,  for  all  percentages  of  neutral,  at  a  ratio  of  air  to  gas  of  5  to  1 ;  but 
they  decrease  as  the  proportion  of  neutral  increases.  The  experiments  indicate 
that  the  speed  of  flame  travel  varies  widely  with  the  nature  of  the  mixture  and  the 
conditions  of  pressure  to  which  it  is  subjected.  If  the  mixture  is  too  weak  or  too 
strong,  it  will  not  inflame  at  all. 

320.  Piston  Speed.  The  actual  shape  of  the  ideally  vertical  ignition  line  will 
depend  largely  upon  the  speed  of  flame  propagation  as  compared  with  the  speed 
of  the  piston.  Figure  139,  after  Lucke,  illustrates  this.  The  three  diagrams  were 
taken  from  the  same  engine  under  exactly  the  same  conditions,  excepting  that  the 
speeds  in  the  three  cases  were  150,  500,  and  750  r.  p.  m.  Similar  effects  may  be 
obtained  by  varying  the  mixture  (and  consequently  the  flame  speed)  while  keep- 
ing the  piston  speed  constant.  High  compression  causes  quick  ignition.  Throt- 


*  The  theoretical  ratio  of  air  to  C6H14  is  47  to  1. 


IGNITION 


173 


tling  of  the  incoming  charge  increases  the  percentage  of  neutral  from  the  burnt 
gases  and  retards  ignition. 


150  r.  p.  m. 


500  r.  p.  m. 


750  r.  p.  m. 

FIG.  139.    Art.  320.  —  Ignition  Line  as  affected  by  Piston  Speed. 
(From  Lucke's  "Gas  Engine  Design.") 

321.  Point  of  Ignition.  The  spreading  of  flame  is  at  first  slow.  Ignition  is, 
therefore,  made  to  occur  prior  to  the  end  of  the  stroke,  giving  a  practically  verti- 
cal line  at  the  end,  where  inflammation  is  well  under  way.  Figure  140,  from 
Poole  (27),  shows  the  effects  of  change  in  the  point  of  ignition.  In  («)  and  (ft), 
ignition  was  so  early  as  to  produce  a  negative  loop  on  the  diagram.  This  was  cor- 
rected in  (e),  but  (cT)  represents  a  still  better  diagram.  In  (e)  and  (/),  ignition 
was  so  late  that  the  comparatively  high  piston  speed  kept  the  pressure  down,  and 
the  work  area  was  small.  It  is  evident  that  too  early  a  point  of  ignition  causes  a 
backward  impulse  on  the  piston,  tending  to  stop  the  engine.  Even  though  the 
inertia  of  the  fly  wheel  carries  the  piston  past  its  "  dead  point,"  a  large  amount  of 
power  is  wasted.  The  same  loss  of  power  follows  accidental  pre-ignition,  whether 
due  to  excessive  compression,  contact  with  hot  burnt  gases,  leakage  past  piston 
rings,  or  other  causes.  Failure  to  ignite  causes  loss  of  capacity  and  irregularity 


174 


APPLIED  THERMODYNAMICS 


of  speed,  but  theoretically  at  least  does  not  affect  economy.  For  reasons  already 
suggested,  light  loads  (where  governing  is  effected  by  throttling  the  supply)  and 
weak  mixtures  call  for  early  ignition. 


IGNITION  25%  EARLY 


GNITION  20%    EARLY 


IGNITION  16%  EARLY 

(c) 


IGNITION  T2%  EARLY 

d 


IGNITION  5%  LATE 


FIG.  140.    Art.  321.  — Time  of  Ignition. 
(From  Poole's  "The  Gas  Engine,"  by  permission  of  the  Hill  Publishing  Company.) 


322.  Methods  of  Ignition.  An  early  method  for  igniting  the  gas  was  to  use 
an  external  flame  enclosed  in  a  rotating  chamber  which  at  proper  intervals  opened 
communication  between  the  flame  and  the  gas.  This  arrangement  was  applicable 
to  slow  speeds  only,  and  some  gas  always  escaped.  In  early  Otto  engines,  the 
external  flame  with  a  sliding  valve  was  used  at  speeds  as  high  as  100  r.  p.  m.  (28). 
The  insertion  periodically  of  a  heated  plate,  once  practiced,  was  too  uncertain. 
The  use  of  an  internal  flame,  as  in  the  Brayton  engine,  was  limited  in  its  applica- 
tion and  introduced  an  element  of  danger.  Self -ignition  by  the  catalytic  action 
of  compressed  gas  upon  spongy  platinum  was  not  sufficiently  positive  and  reliable. 
The  use  of  an  incandescent  wire,  electrically  heated  and  mechanically  brought 
into  contact  with  the  gas,  was  a  forerunner  of  modern  electrical  methods.  The 
"hot  tube"  method  is  still  in  frequent  use,  particularly  in  England.  This  in- 
volves the  use  of  an  externally  heated  refractory  tube,  which  is  exposed  to  the  gas 
either  intermittently  by  means  of  a  timing  valve,  or  continuously,  ignition  being 
then  controlled  by  adjusting  the  position  of  the  external  flame.  In  the  Hornsby- 
Akroyd  and  Diesel  engines,  ignition  is  self-induced  by  compression  alone ;  but 
external  heating  is  necessary  to  start  these  engines. 


IGNITION  AND  EXPANSION 


175 


323.  Electrical  Methods.     The  two  modern  electrical  methods  are  the 
"  make  and  break  "  and  "  jump  spark.''     In  the  former,  an  electric  current, 
generated  from  batteries  or  a  small  dynamo,  is  passed  through  two  sepa- 
rable contacts  located  in  the  cylinder  and  connected  in  series  with  a  spark 
coil.     At  the  proper  instant,  the  contacts  are  separated  and  a  spark  passes 
between  them.     In  the  jump  spark  system,  an  induction  coil  is  used  and 
the  contacts  are  stationary.  A  series  of  sparks  is  thrown  between  them  when 
the  primary  circuit  is  closed,  just  before  the  end  of  the  compression  stroke. 

324.  Clearance   Space.     The   combustion   chamber  formed  in  the  clearance 
space  must  be  of  proper  size  to  produce  the  desired  final  pressure.     A  common 
ratio  to  piston  displacement  is  30  per  cent.     Hutton  has  shown    (29)  that  the 
limits  for  best  results  may  range  easily  from  8.7  to  56  per  cent  (Art.  332) . 

325.  Expansion  Curve.     Slow  inflammation  has  been  shown  to  result  in  a  de- 
creased maximum  pressure  after  ignition.     Inflammation  occurring  during  expan- 
sion as  a  result  of  slow  spreading  of  the  flame  is  called  "  after  burning"     Ideally, 
the  expansion  curve  should  be  adiabatic ;  actually  it  falls  in  most  cases  above  the 
air  adiabatic,  pv1-40'2  =  constant,  although  it  is  known  that  during  expansion  from  40 
to  50  per  cent  of  the  total   heat   in   the   gas   is   being 

carried  away  by  the  jacket  water.  Figure  141  repre- 
sents an  extreme  case;  after  burning  has  made  the 
expansion  line  almost  horizontal,  and  some  unburnt 
gas  is  being  discharged  to  the  exhaust.  Those  who 
hold  to  the  dissociation  theory  would  explain  this 
line  ou  the  ground  that  the  gases  dissociated  during  combustion  are  gradually 
combining  as  the  temperature  falls ;  but  actually,  the  temperature  is  not  falling, 
and  the  effect  which  we  call  after  burning  is  most  pronounced  with  weak  mix- 
tures and  at  such  low  temperatures  as  do  not  permit  of  any  considerable 
amount  of  dissociation.  Practically,  dissociation  has  the  same  effect  as  an 
increasing  specific  heat  at  high  temperature.  It  affects  the  ignition  line  to 
some  extent ;  but  the  shape  of  the  expansion  line  is  to  a  far  greater  de- 
gree determined  by  the  slow  inflammation  of  the  gases.  The  effect  of 
the  transfer  of  heat  between  the  fluid  and  the  cylinder  walls  is  dis- 
cussed in  Art.  347.  The  actual  exponent  of.  the  expansion 
curve  varies  from  1.2  in  large  engines  to  1.38  in  good  small 
engines,  occasionally,  however,  rising  as  high  as  1.5. 
The  compression  curve  usually,  though 
not  always,  has  a  slightly 
higher  exponent.  The 
adiabatic  exponent  for  a 

FIG,  142.    Art.  325. -Explosion  Waves.  mixture    of    hydrocarbon 

gases  is  lower  than  that 

for  air  or  a  perfect  gas ;  and  in  some  cases  the  actual  adiabatic,  plotted  for  the 
gases  used,  would  be  above  the  determined  expansion  line,  as  should  normally  be 
expected,  in  spite  of  after  burning.  The  presence  of  explosion  waves  (Art,  319) 
may  modify  the  shape  of  the  expansion  curve,  as  in  Fig.  142.  The  equivalent 


FIG. 


141.      Art.    325.  —  After 
Burning. 


176 


APPLIED  THERMODYNAMICS 


curve  may  be  plotted  as  a  mean  through  the  oscillations.     Care  must  be  taken 
not  to  confuse  these  vibrations  with  those  due  to  the  inertia  of  the  indicating 

instrument. 


326-  The  Exhaust  Line.  This  is 
shown  to  an  enlarged  vertical  scale 
as  be,  Fig.  137.  - "  Low  spring  "  dia- 
grams of  this  form  are  extremely  use- 
ful. As  engines  wear,  more  or  less 
"  lost  motion  "  becomes  present  in  the 


FIG.  143.    Art.   326.  — Delayed  Exhaust  Valve 
Opening. 


valve-actuating  gear,  and  the  tendency  of  this  is  to  vary  the  instant  of  opening 
or  closing  the  inlet  or  the  exhaust  valve.  The  effect  of  delayed  opening  of  the 
latter  is  shown  in  Fig.  143 ;  that 
of  an  inadequate  exhaust  passage, 
in  Fig.  144.  An  early  opening 
of  the  exhaust  valve  may  cause 
loss  also,  as  in  Fig. -145.  In  mul- 
tiple cylinder  engines  having  corn- 


mon  exhaust  and  suction  mains, 


FIG.  144.    Art.  326.  —  Throttled  Exhaust  Passages. 


early  exhaust  from  one  cylinder 
may  produce  a  rise  of  pressure  during  the  latter  part  of  the  exhaust  stroke  of 
another.  Obstructions  to  suction  and  discharge  movements  of  gas  are  com- 
monly classed  together  as 
"  fluid  friction."  This  may  in 
small  engines  amount  to  as 
much  as  30  per  cent  of  the 
power  developed.  In  good 
engines  of  large  or  moderate 
size,  it  should  not  exceed  6  per 
cent.  It  increases,  propor- 
tionately, at  light  loads  ;  and 


FIG.  145.    Art.  326.  —  Exhaust  Valve  Opening  too  Early. 


possibly  absolutely  as  well  if  governing  is  effected  by  throttling  the  charge. 

327.  Scavenging.  To  avoid  the  presence  of  burnt  gases  in  the  clear- 
ance space,  and  their  subsequent  mingling  with  the  fresh  charge,  "scav- 
enging," or  sweeping  out  these  gases  from  the  cylinder,  is  sometimes  prac- 
ticed. This  may  be  accomplished  by  means  of  a  separate  air  pump,  or  by 
adding  two  idle  strokes  to  the  four  strokes  of  the  Otto  cycle.  In  the 
Crossley  engines,  the  air  admission  valve  was  opened  before  the  gas  valve, 
and  before  the  termination  of  the  exhaust  stroke.  By  using  a  long  ex- 
haust pipe,  the  gases  were  discharged  in  a  rather  violent  puff,  which  pro- 
duced a  partial  vacuum  in  the  cylinder.  This  in  turn  caused  a  rush  of 
air  into  the  clearance  space,  which  swept  out  the  burnt  gases  by  the  time 
the  piston  had  reached  the  end  of  its  stroke.  Scavenging  decreases  the 
danger  of  missing  ignitions  with  weak  gas,  tends  to  prevent  pre-ignition, 
and  appears  to  have  reduced  the  consumption  of  fuel. 


DIAGRAM   FACTOR 


177 


328.  The  Suction  Stroke.  This  also  is  shown  in  Fig.  137,  line  cd.  The  effect 
of  late  opening  of  the  valve  is  shown  in  Fig.  146 ;  that  of  an  obstructed  passage 
or  of  throttling  the  supply,  in  Fig. 
147.  If  the  opening  is  too  early, 
exhaust  gases  will  enter  the  supply 
pipe.  If  closure  is  too  early,  the 
gas  will  expand  during  the  re- 
mainder of  the  suction  stroke,  but 
the  net  work  lost  is  negligible;  if 
too  late,  some  gas  will  be  discharged 
back  to  the  supply  pipe  during  the 
beginning  of  the  compression  stroke, 
as  in  Fig.  148.  Excessive  obstruc- 
tion, in  the  suction  passages  de- 
creases the  capacity  of  the  engine, 
in  a  way  already  suggested  in  the 
study  of  air  compressors  (Art.  224). 


FIG.  146.    Art.  328.  —  Delayed  Opening  of 
Suction  Valve. 


FIG.  147.    Art. 


ACTUAL 

J.  —  Throttled  Suction. 


FIG.  148.    Art.  328.  —  Late  Closing  of 
Suction  Valve. 


329.    Diagram  Factor.    The 

discussion  of  Art.  309  to  Art. 
328  serves  to  show  why  the 
work  area  of  any  actual  dia- 
gram must  always  be  less  than 
that  of  the  ideal  diagram  for 
the  same  cylinder,  as  given  in 
Fig.  122.  The  ratio  of  the 
two  is  called  the  diagram 

factor.  The  area  of  the  ideal  card  would  constantly  increase  as 
compression  increased  ;  that  of  the  actual  card  soon  reaches  a  limit 
in  this  respect;  and,  consequently,  in  general,  the  diagram  factor 
decreases  as  compression  increases.  Variations  in  excellence  of 
design  are  also  responsible  for  variations  of  diagram  factor. 

In  the  best  recorded  tests,  its  value  has  ranged  from  0.38  to  0.59 ;  in 
ordinary  practice,  the  values  given  by  Lucke  (30)  are  as  follows:  for 
kerosene,  if  previously  vaporized  and  compressed,  0.30  to  0.40,  if  injected 
on  a  hot  tube,  0.20;  for  gasoline,  0.52  to  0.50;  for  producer  gas,  0.40  to 
0.56;  for  coal  gas,  0.45 ;  for  carbureted  water  gas,  0.45;  for  blast  furnace 
gas,  0.30  to  0.48;  for  natural  gas,  0.40  to  0.52.  These  figures  are  for  four- 
cycle engines.  For  two-cycle  engines,  usual  values  are  about  20  per  cent 
less.  Figure  149  shows  on  the  PV  and  entropy  planes  an  actual  indicator 
diagram  with  the  corresponding  ideal  cycle. 


178 


APPLIED  THERMODYNAMICS 


ACTUAL  DIAGRAM 
IDEAL  DIAGRAM 


FIG.  149.    Art.  329.  —  Actual  and  Ideal  Gas  Engine  Diagrams. 


GAS  ENGINE  DESIGN 

330.  Capacity.     The  work  done  per  stroke  may  readily  be  computed  for  the 
ideal  cycle,  as  in  Art.  293.     This  may  be  multiplied  by  the  diagram  factor  to 
determine  the  probable  performance  of   an  actual  engine.     To  develop  a  given 
power,  the  number  of  cycles  per  minute  must  be  established.     Ordinary  piston 
speeds  are  from  450  to  1000  ft.  per  minute,  usually  lying  between  550  and  800  ft., 
the  larger  engines  having  the  higher  speeds.     The  stroke  ranges  from  1.0  to  2.0 
times  the  diameter,  the  ratio  increasing,  generally,  with  the  size  of  the  engine. 
A  gas  engine  has  no  overload  capacity,  strictly  speaking,  since  all  of  the  factors 
entering  into  the  determination  of  its  capacity  are  intimately  related  to  its  effi- 
ciency.    It  can  be   given   a  margin  of  capacity  by  making  it  larger  than  the 
computations  indicate  as  necessary,  but  this  or  any  other  method  involves  a  con- 
siderable sacrifice  of  the  economy  at  normal  load. 

331.  Mean  Effective  Pressure.     Since  in  an  engine  of  given  size  the  extreme 
volume  range  of  the  cycle  is  fixed,  the  mean  net  ordinate  of  the  work  area  measures 
the  capacity.    The  quotient  of  the  cycle  area  by  the  volume  range  gives  what  is  called 
the  mean  effective  pressure  (m.  e.  p.).     In  Fig.  122,  it  is  ebfd  +  (Vd  —  F,).     We 

y-i 

then  write  m.  e.  p.  =  W  +  (  Vd  -  F.);  but  from  Art.  295,  W  =  Q[I  -  (—  )  *  ]  ;    Q 

being  the  gross  quantity  of  heat  absorbed  in  the  cycle.     Then,  in  proper  units, 
without  allowance  for  diagram  factor, 


m.  e.  p.  = 


Vd  -  V. 


332.    Illustrative  Problem.     To  determine  the  cylinder  dimensions  of  a  four-cycle^ 
two-cylinder,  double-acting  engine  of  500  hp.,  using  producer  gas  (assumed  to  contain 


GAS   ENGINE   DESIGN 


179 


CO,  39.4;  N,  60;  H,  0.6;  parts  in  100  by  weight)  (Art.  285),  at  150  r.  p.  m.  and  a 
piston  speed  of  825  ft.  per  minute. 

We  assume  (Fig.  150),  PL  =  12,  P2  =  144.7,  Tl  =  200°  F.,  and  diagram  factor 
=  0.48  (Arts  312,  329). 


=  5.9.     Let  the  piston  displace- 


n 
Since  Pl  l\»  =  Pz  Vj>,  ~*  =    jf     = 


ment  F,  -  V*  =  D.     Then   F2  =  0.2045  D  and  F,  =  1.2045  D.     The  clearance  is 
£  =  02043    (Art.    324)».      Also     7%  =  ^p  =  «2Mj|i«^p45  =  ^ 

absolute.  The  heat  evolved  per  pound  of  the  mixed  gas  (taking  the  calorific 
value  of  hydrogen  burned  to  steam  as  53,400)  is  (0.394  x  4315) +  (0.006  x  53,400) 
=  2021  B.  t.  u.  The  products  of  com- 
bustion consist  of  ||  x  0.394  =  3 
0.619  Ib.of  CO2  (specific  heat= 0.1692), 
0.006  x  9  =  0.054  Ib.  of  H2O  (steam, 
specific  heat  0.37),  and  ||  (0.619  - 
0.394)  =  0.751  Ib.  of  X  accompanying 
the  oxygen  introduced  to  burn  the 
CO,  with  (0.054-0.006)H  =  0.1607  Ib. 
of  X  accompanying  the  oxygen  in- 
troduced to  burn  the  II ;  and  0.60  Ib. 
of  X  originally  in  the  gas,  making  a 
total  of  1.5117  Ib.  of  X  (specific  heat 
0.1727).  To  raise  the  temperature  of 
these  constituents  1°  F.  at  constant 


FIG.  150.    Arts.  332-335.  —  Design  of  Gas 
Engine. 


volume  requires  (0.619  x  0.1692)  +  (0.054x0.37)  -f  (1.5117  x  0.1727)  =  0.3849 
B.  t.  u.  The  rise  in  temperature  T3  -  T*  is  then  2021  -=-  0.3849  =  5260°,  and 
Ta  =  5260  +  1357  =  6617°  absolute.  Then 


and 

The  work  per  cycle  is 


-144x0  48  j  x  0.2045) -(58.7xl.2045) -(144.7  x  0.2045)  +  (12  x  1.2045)1 

0.402 
=  10,080  D  foot  pounds. 

In  a  two-cylinder,  four-cycle,  double-acting  engine,  all  of  the  strokes  are  work- 
ing strokes ;  the  foot-pounds  of  work  per  stroke  necessary  to  develop  500  hp.  are 

*  While  the  use  of  a  "  blanket  • '  diagram  factor  as  in  this  illustration  may  be  justi- 
fied, in  any  actual  design  the  clearance  at  least  must  be  ascertained  from  the  actual 
exponent  of  the  compression  curve.  The  design  as  a  whole,  moreover,  would  better 
be  based  on  special  assumptions  as  in  Problem  15,  (ft),  page  197. 


180  APPLIED  THERMODYNAMICS 

QQfinO  v    KAA 

01--  55>000-      The    necessary    piston    displacement    per    stroke,    D,    is 
2  x  loO 

55,000  -  10,080  =  5.46  cu.  ft.  The  stroke  is  825  -j-  (2  x  150)  =  2.75  ft.  or  33  in.  The 
piston  area  is  then  5.  46  -=-2.  75  =  1.985  sq.  ft.  or  285.5  sq.  in.  The  area  of  the  water- 
cooled  tail  rod  rnay  be  about  33  sq.  in.,  so  that  the  cylinder  area  should  be 
285.5  +  33  =  318.5  sq.  in.  and  its  diameter  consequently  20.14  in. 

333.  Modified  Design.  In  an  actual  design  for  the  assumed  conditions,  over- 
load capacity  was  secured  by  assuming  a  load  of  600  hp.  to  be  carried  with  20  per 
cent  excess  air  in  the  mixture.  (At  theoretical  air  supply,  the  power  developed 
should  then  somewhat  exceed  600  hp.)  The  air  supply  per  pound  of  gas  is  now 

[(0.394  x  |f)  +  (0.006  x  8)]  -\^  x  1  .2  =  1.422  Ib. 

Of  this  amount,  0.23  x  1.422  =  0.327  Ib.  is  oxygen.  The  products  of  combustion 
are  f  f  x  0.394  =  0.619  Ib.  CO2,  0.006  x  9  =  0.054  Ib.  H2O,  (1.422  -  0.327)  +  0.60 
=  1.693  Ib.  N,  and  0.327-  Q|  x  0.394)  -  (8  x  0.006)  =  0.054  Ib.  of  excess  oxygen  ;  a 
total  of  2.422  Ib.  The  rise  in  temperature  T3-  T2  is 

9001 

-  £2*  _  __  =  4700°. 
(0.619  x  0.1692)  +  (0.054  x  0.37)  +  (1.693  x  0.1727)  +  (0.054  x  0.1551) 

Then  !T3  =  4760  +  1357  =  61  17°  absolute, 


and  the  work  per  cycle  is 

144  x  0  48  £>r(655  x  0.2045)  -  (54.2  x  1.2045)  -  (144.7  x  0.2045)  +  (12  x  1.2045)1 

0.402  J 

=  9150  D  foot-pounds. 

The  piston  displacement  per  stroke  is    60°  *  330QQ    —  7.21   cu.   ft.,  the  cylinder 

2  x  150  x  9150 

area  is  (7.21  -f-  2.75)  144  +  33  =  410  sq.  in.,  and  its  diameter  22.83  in.     The  cylinders 
were  actually  made  23  &  by  33  in.,  the  gas  composition  being  independently  assumed. 

334.  Estimate  of  Efficiency.  To  determine  the  probable  efficiency  of  the  engine 
under  consideration  :  each  pound  of  working  substance  is  supplied  with  1.422  Ib. 
of  air.  Multiplying  the  weights  of  the  constituents  by  their  respective  specific 
volumes,  we  obtain  as  the  volume  of  mixture  per  pound  of  gas,  31.33  cu,  ft.  at 
14.7  Ib.  pressure  and  32°  F.,  as  follows  :  — 

CO,  0.394  x    12.75    =  5.01 

H,  0.006  x  178.83    =   1.07 

N.  0.600  x    12.75    =  7.65 

Air,  1.422  x    12.387  =17.60 

31.33 

At  the  state  1,  Fig.  150,  7\  =  659.6,  P^  =  12,  whence 

y  =  T^PnVn  =  659.6  x  14.7x31.33  =  „  2 
1       P,ro  12  x  491.6 

The  piston  displaces  7.21  x  300  =  2163  cu.  ft.  or  2163  -*-  51.2  =  42.3  Ib.  of  this  mix- 
ture per  minute.     The  heat  taken  in  per  minute  is  then  2021  x  42.3  =  85,200  B.  t.  u. 


AUTOMOBILE   ENGINE  RATING  181 


fiOO  v 

The  work  done  per  minute  is  —     x  aouuu  _  %5500  B  f   M>      The  e^cjency  js 

778 
25,500  -;-  85,200  =  0.299.     An  actual  test  of  the  engine  gave  0.282,  with  a  load 

somewhat  under  600  hp.     The  Otto  cycle  efficiency  is  18o7~6°9-6  =  0.516. 

1357 

335.  Automobile  Engine.  To  ascertain  the  probable  capacity  and  economy  of  a 
four-cylinder,  four-cycle,  single-acting  gasoline  engine  with  cylinders  4-  by  5  in.,  at 

1500  r.  p.  m. 

In  Fig.  150,  assume  P,  =  12,  P2  =  84.7,  Tl  =  70°  F.,  diagram  factor,  0.375 
(Arts.  312,  329).  Assume  the  heating  value  of  gasoline  at  19,000  B.  t.  u.,  and  its 
composition  as  C6H14  :  its  vapor  density  as  3.05  (air  =  1.).  Let  the  theoretically 
necessary  quantity  'of  air  be  supplied. 

The  engine  will  give  two  cycles  per  revolution.     Its  active  piston  displacement 


is  then  °'7854  x  (4*)2  x  °  x  3000  =  145.5  cu.  ft.  per  minute,  which  may  be  repre- 

sented as  Vl  -  Vy  Fig.  150.     We  now  find 

F,  =  /  J12_W13  =  Q  2495  .   72  =  0.2495  Vi  ;  0.7505  Vl  =  145.5  ;   Vl  =  194  ;  F2=48.5  ; 


Clearance  =  =  0.334  (Art.  324)  ;   T2  =  84J  '*  -  936°  absolute. 


To  burn  one  pound  of  gasoline  there  are  required  3.53  Ib.  of  oxygen,  or  15.3  Ib. 
of  air.  For  one  cubic  foot  of  gasoline,  we  must  supply  3.05  x  15.3  ==  46.6  cu.  ft. 
of  air.  The  145.5  cu.  ft.  of  mixture  displaced  per  minute  must  then  consist  of 
145.5  -5-  47.6  =  3.06  cu.  ft.  of  gasoline  and  142.44  cu.  ft.  of  air,  at  70°  F.  and  12  Ib. 

pressure.     The  specific  volume  of  air  at  this  state  is  5'29'6  x  14'7  x  12-387  =  16.38 

cu.  ft.  ;  that  of  gasolene  is  16.38  -f-  3.05  =  5.37  cu.  ft.  The  weight  of  gasoline 
used  per  minute  is  then  3.06  -f-  5.37  =  0.571  Ib.  The  heat  used  per  minute  is 
0.571  x  19,000  =  10,840  B.  t.  u.  The  combustion  reaction  may  be  written 

C6H14  +  O19  =  6  CO2  +  7  H2O 
86  +  304  =  264  +  126 

V£  =    3.06  Ib.  C02  per  Ib.  C6H14 
¥£•  =    1-35  Ib.  H2O  per  Ib.  C6H14 
H  x  W  =  11-82  Ib.  N  per  Ib.  C6H14 

16.23  =  1.  +  15.3,  approximately. 

The  heat  required  to  raise  the  temperature  of  the  products  of  combustion  1°  F.  is 
[(3.06x0.1692)  +  (1.35  x  0.37)  +  (11.82  x  0.1727)]  0.571  =  1.646  B.  t.  u.  per  minute. 
The  rise  in  temperature  T*  -  T.2  is  then  10,840  -*-  1.646  =  6610°,  Tz  =  6610  +  936 


=  7546°  absolute,  P3  =  84.7          =  681  ,  P4  =  12  —  =  96.7,  and  the  work  per  minute  is 

«7OU  Oi.  / 

0  375  x  144  ["(681  x  48.5)-  (96.7  x  194)  -  (84.7  x  48.5)  +  (12  x  194)  1  =  l  6g2  m 

L  0.402 

foot-pounds.      This  is   equivalent   to    lAff^a*  =  2160  B.  t.  u.  per  minute   or   to 
=  51  horsepower.     The  efficiency  is  2160  -4-  10840  =  0.00.     In  an  automobile 


182 


APPLIED  THERMODYNAMICS 


running  at  50  miles  per  hour,  this  would  correspond  to  50^-  (0.571  x  60)  —  1.46  miles 
run  per  pound  of  gasoline.  In  practice,  the  air  supply  is  usually  deficient,  and  the 
power  and  economy  less  than  those  computed. 

It  is  obvious  that  with  a  given  fuel,  the  diagram  factor  and  other  data  of 
assumption  are  virtually  fixed.  An  approximation  of  the  power  of  the  engine 
may  then  be  made,  based  on  the  piston  displacement  only.  This  justifies  in  some 
measure  the  various  rules  proposed  for  rating  automobile  engines  (30  a). 

CURRENT  GAS  ENGINE  FORMS 

336.  Otto  Cycle  Oil  Engines.  This  class  includes,  among  many  others,  the 
Mietz  arid  Weiss,  two-cycle,  and  the  Daimler,  Priestmaii,  and  Hornsby-Akroyd, 
four-cycle.  In  the  last  named,  shown  in  Fig.  151,  kerosene  oil  is  injected  by  a 


FIG.  151.    Art.  338.  —  Kerosene  Engine  with  Vaporizer. 
(From  "The  Gas  Engine,"  by  Cecil  P.  Poole,  with  the  permission  of  the  Hill  Publishing  Company.) 


small  pump  into  the  vaporizer.  Air  is  drawn  into  the  cylinder  during  the  suction 
stroke,  and  compressed  into  the  vaporizer  on  the  compression  stroke,  where  the 
simultaneous  presence  of  a  critical  mixture  and  a  high  temperature  produces  the^ 
explosion.  External  heat  must  be  applied  for  starting.  The  point  of  ignition  is 
determined  by  the  amount  of  compression  ;  and  this  may  be  varied  by  adjusting 
the  length  of  the  connecting  rod  on  the  valve  gear.  The  engine  is  governed  by 
partially  throttling  the  charge  of  oil,  thus  weakening  the  mixture  and  the  force 
of  the  explosion.  The  oil  consumption  may  be  reduced  to  less  than  1  Ib.  per 
brake  hp.  per  hour. 


TYPES   OF  GAS  ENGINE 


183 


In  the  Priestman  engine,  an  earlier  type,  air  under  pressure  sprayed  the  oil 
into  a  vaporizer  kept  hot  by  the  exhaust  gases.  The  method  of  governing  was  to 
reduce  the  quantity  of  charge  without  changing  its  proportions.  A  hand  pump 
and  external  heat  for  the  vaporizer  were  necessary  in  starting.  An  indicated 
thermal  efficiency  of  0.165  has  been  obtained.  The  Daimler  (German) ^engine 
uses  hot-tube  ignition  without  a  timing  valve,  the  hot  tube  serving  as  a  vaporizer. 
Extraordinarily  high  speeds  are  attained. 

337.  Modern  Gas  Engines  :  the  Otto.  The  present-day  small  Otta^ngine^is  ordi- 
narily single-cylinder  and  single-acting,  governing  on  the  "  hjt  or  miss  "  principle 
(Art.  343).  It  is  used  with  all  kinds  of  gas  and  with"  gasoline.  Ignition  is  elec- 
trical, the  cylinder  water  jacketed,  the  jackets  cast  separately  from  the  cylinder. 
The  Foos  engine,  a  simple,  compact  form,  often  made  portable,  is  similar  in  princi- 
ple. In  the  Crossley-Otto,  a  leading  British  type,  hot-tube  ignition  is  used,  and 
the  large  units  have  two  horizontal  opposed  single-acting  cylinders.  In  the 
Andrews  form,  tandem  cylinders  are  used,  the  two  pistons  being  connected  by 
external  side  rods. 


338.  The  Westinghouse  Engine.  This  has  rectently  been  developed  in  very 
large  units.  Figure  152  shows  the  working  side  of  a  two-cylinder,  tandem, 
double  acting  engine,  representing  the  inlet  valves  on  top  of  the  cylinders. 


FIG.  152.    Arts.  338,  350.  —  Westinghouse  Gas  Engine.    Two-cylinder  Tandem,  Four-cycle. 

Smaller 'engines  are  often  built  vertical,  with  one,  two,  or  three  single-acting 
cylinders.  All  of  these  engines  are  four-cycle,  with  electric  ignition,  governing 
by  varying  the  quantity  and  proportions  of  the  admitted  mixture.  Sections  of 
the  cylinder  of  the  Riverside  horizontal,  tandem,  double-acting  engine  are  shown  in 
Fig.  153.  It  has  an  extremely  massive  frame.  The  Allis-Chalmers  engine  is  built 
in  large  units  along  similar  general  lines.  Thirty-six  of  the  latter  engines  of 
4000  hp.  capacity  each  on  blast  furnace  gas  are  now  (1909)  being  constructed. 
They  weigh,  each,  about  1,500,000  lb.,  and  run  at  83 l  r.  p.  m.  The  cylinders  are 
44  by  54  in.  Nearly  all  are  to  be  direct-connected  to  electric  generators. 


184 


APPLIED  THERMODYNAMICS 


TYPES  OF  GAS  ENGINE  185 

339.  Two-cycle  Engines.     In  these,  the  explosions  are  twice  as  frequent  as 
with  the  four-cycle  engine,  and  cooling  is  consequently  more  difficult.     With  an 
equal  number  of  cylinders,  single-  or  double-acting,  the  two-cycle  engine  of  course 
gives  better  regulation.     The  first  important  two-cycle  engine  was  introduced  by 
Clerk  in  1880.    The  principle  was  the  same  as  that  of  the  engine  shown  in  Fig.  119. 
The  Oechelhaueser  engine  has  two  single-acting  pistons  in  one  cylinder,  which  are 
connected  with  cranks  at  180°,  so  that  they  alternately  approach  toward  and 
recede  from  each  other.     The  engine  frame  is  excessively  long.     Changes  in  the 
quantity  of  fuel  supplied  control  the  speed.     The  Koerting  engine,  a  double-acting 
horizontal  form,  has  two  pumps,  one  for  air  and  one  for  gas.     A  "  scavenging  " 
charge  of  air  is  admitted  just  prior  to  the  entrance  of  the  gas,  sweeping  out  the 
burnt  gases  and  acting  as  a  cushion  between  the  incoming  charge  and  the  exhaust 
ports.     The  engine  is  built  in  large  units,  with  electrical  ignition  and  compressed 
air  starting  gear.     The  speed  is  controlled  by  changing  the  mixture  proportions. 

340.  Special  Engines.     For  motor  bicycles,  a  single  air-cooled  cylinder  is  often 
used,  with  gasoline  fuel.     Occasionally,  two  cylinders  are  employed.     The  engine 
is  four-cycle  and  runs  at  high  speed.     Starting  is  effected  by  foot  power,  which 
can  be  employed  whenever  desired.     Ignition  is  electrical  and  adjustable.     The 
speed  is  controlled  by  throttling.     Extended  surface  air-cooled  cylinders  have  also 
been  used  on  automobiles,  a  fan  being  employed  to  circulate  the  air,  but  the  limit 
of  size  appears  to  be  about  7  hp.  per  cylinder.     Most  automobiles  have  water- 
cooled  cylinders,  usually  four  in  number,  four-cycle,   single-acting,  running  at 
about  1000  to  1200  r.  p.  m.,  normally.     Governing  is  by  throttling  and  by  chang- 
ing the  point  of  ignition.     The  cylinders  are  usually  vertical,  the  jacket  water 
being  circulated  by  a  centrifugal  pump,  and  being  used  repeatedly.     Both  hot-tube 
and  electrical  methods  of  ignition  have  been  employed,  but  the  former  is  now 
almost  wholly  obsolete.     The  number  of  cylinders  varies  from  one  to  six ;  occa- 
sionally they  are  arranged  horizontally,  duplex,  or  opposed.     Two-cycle  engines 
have  been  introduced.     The  fuel  in  this  country  is  usually  gasoline.     For  launch 
engines,  the  two-cycle  principle  is  popular,  the  crank  case  forming  the  pump 
chamber,  and  governing  being  accomplished  by  throttling.     Kerosene  or  gasoline 
are  the  fuels. 

341.  Alcohol  Engines.     These  are  used  on  automobiles  in  France.     A  special 
carburetor  is  employed.     The  cylinder  and  piston  arrangement  is  sometimes  that 
of  the  Oechelhaueser  engine  (Art.  339).     The  speed  is  controlled  by  varying  the 
point  of  ignition.     In  launch  applications,  the  alcohol  is  condensed,  on  account  of 
its  high  cost,  and  in  some  cases  is  not  burned,  but  serves  merely  as  a  working  fluid 
in  a  "  steam  "  cylinder,  being  alternately  vaporized  by  an  externally  applied  gaso- 
line flame  and  condensed  in  a  surface  condenser.     The  low  value  of  the  latent 
heat  of  vaporization  (Art.  360)  of  alcohol  permits  of  " getting  up  steam"  more 
rapidly  than  is  possible  in  an  ordinary  steam  engine. 

342.  Basis  of  Efficiency.     The  performances  of  gas  engines  may  be  compared 
by  the  cubic  feet  of  gas,  or  pounds  of  liquid  fuel,  or  pounds  of  coal  gasified  in  the 
producer,  per  horse  power  hour ;  but  since  none  of  these  figures  affords  any  really 


186  APPLIED  THERMODYNAMICS 

definite  basis,  on  account  of  variations  in  heating  value,  it  is  usual  to  express  the 
results  of  trials  in  heat  units  consumed  per  horse  power  per  minute.  Since  one  horse 
power  equals  33,000  H-  778  =  42.42  B.  t.  u.  per  minute,  this  constant  divided  by  the 
heat  unit  consumption  gives  the  indicated  thermal  efficiency.  In  making  tests,  the 
over-all  efficiency  of  a  producer  plant  may  be  ascertained  by  weighing  the  coal. 
When  liquid  fuel  is  used,  the  engine  efficiency  can  readily  be  determined  separately. 
To  do  this  with  gas  involves  the  measurement  of  the  gas,  always  a  matter  of  some 
difficulty  with  auy  but  small  engines.  The  measurement  of  power  by  the  indicator 
is  also  inaccurate,  possibly  to  as  great  an  extent  as  5  per  cent,  which  may  be  reduced  to 
2  per  cent,  according  to  Hopkinson,  by  employing  mirror  indicators.  This  error  has 
resulted  in  the  custom  of  expressing  performance  in  heat  units  consumed  per  brake 
horse  power  per  hour  or  per  kw.-hr.,  where  the  engines  are  directly  connected  to 
generators.  There  is  some  question  as  to  the  proper  method  of  considering  the 
negative  loop,  bcde,  of  Fig.  136.  By  some,  its  area  is  deducted  from  the  gross  work 
area,  and  the  difference  used  in  computing  the  indicated  horse  power.  By  others, 
the  gross  work  area  of  Fig.  136  is  alone  considered,  and  the  "  fluid  friction  "  losses 
producing  the  negative  loop  are  then  classed  with  engine  friction  as  reducing  the 
"  mechanical  efficiency."  Various  codes  for  testing  gas  engines  are  in  use  (31). 

343.  Typical  Figures.     Small  oil  or  gasoline  engines  may  easily  show  10  per 
cent  brake  efficiency.     Alcohol  engines  of  small  size  consume  less  than  2  pt.  per 
brake  hp.-hr.  at  full  load  (32).     A  well-adjusted  Otto  engine  has  given  an  indicated 
thermal  efficiency  of  0.19  with  gasoline  and  0.23  with  kerosene  (33).     Ordinary 
power  gas  engines  of  average  size  under  test  conditions  have  repeatedly  shown 
indicated  thermal  efficiencies  of  25  to  29  per  cent.     A  Cockerill  engine  gave  30  per 
cent.     Hubert  (34)  tested  at  Seraing  an  engine  showing  nearly  32  per  cent  indicated 
thermal  efficiency.     Mathot  (35)  reports  a  test  of  an  Ehrhardt  and  Lehmer  double- 
acting,  four-cycle  600  hp.  engine  at  Heinitz  which  reached  nearly  38  per  cent.     A 
blast  furnace  gas  engine  gave  at  full  load  25.4  per  cent.     Expressed  in  pounds  of 
coal,  one  plant  with  a  low  load  factor  gave  a  kilowatt-hour  per  1.8  Ib.     In  another 
case,  1.59  Ib.  was  reached,  and  in  another,  2.97  Ib.  of  wood  per  kw.rhr.     It  is  common 
to  hear  of  guarantees  of  1  Ib.  of  coal  per  brake  hp.-hr.,  or  of  11,000  B.  t.  u.  in  gas. 
A  recent  test  of  a  Crossley  engine  is  reported  to  have  shown  the  result  1.13  Ib.  of 
coal  per  kw.-hr.     Under  ordinary  running  conditions,  1.5  to  2.0  Ib.  with  varying 
load  may  easily  be  realized.     These  latter  figures  are  of  course  for  coal  burned  in 
the  producer.     They  represent  the  joint  efficiency  of  the  engine  and  the  producer. 
The  best  results  have  been  obtained  in  Germany.     For  the  engine  alone,  Schroter 
is  reported  to  have  obtained  on  a  Guldner  engine  an  indicated  thermal  efficiency  of 
0.427  at  full  load  with  illuminating  gas  (36). 

The  efficiency  cannot  exceed  that  of  the  ideal  Otto  cycle.  In  one  test  of  an 
Otto  cycle  engine  an  indicated  thermal  efficiency  of  0.37  was  obtained,  while  the 
ideal  Otto  efficiency  was  only  0.41.  The  engine  was  thus  within  10  per  cent  of 
perfection  for  its  cycle. 

The  Diesel  engine  has  given  from  0.32  to  0.412  indicated  thermal  efficiency. 
Its  cycle,  as  has  been  shown,  permits  of  higher  efficiency  than  that  of  Otto. 

344.  Plant  Efficiency.     Figures  have  been  given  on  coal  consumption.     Over- 
all efficiencies  from  fuel  to  indicated  work  have  ranged  from  0.1 4  upward.     At  the 


GAS  ENGINE  TRIALS  187 

Maschinenfabrik  Winterthur,  a  consumption  of  0.7  Ib.  of  coal  (13,850  B.  t.  u.)  per 
brake  hp.-hr.  at  full  load  has  been  reported  (37).  This  is  closely  paralleled  by  the 
0.285  plant  efficiency  obtained  on  the  Guldner  engine  mentioned  in  Art.  343  when 
operated  with  a  suction  producer  on  anthracite  coal.  At  the  Royal  Foundry, 
Wurtemburg  (38),  0.78  Ib.  of  anthracite  were  burned  per  Ihp.-hr.,  and  at  the 
Imperial  Post  Office,  Hamburg,  0.93  Ib.  of  coke.  In  the  best  engines,  variations  of 
efficiency  with  reasonable  changes  of  load  below  the  normal  have  been  greatly 
reduced,  largely  by  improved  methods  of  governing. 

345.  Mechanical  Efficiency.  The  ratio  of  .work  at  the  brake  to  net  indicated 
work  ranges  about  the  same  for  gas  as  for  steam  engines  having  the  same  arrange- 
ment of  cylinders.  When  mechanical  efficiency  is  understood  in  this  sense,  its 
value  is  nearly  constant  for  a  given  engine  at  all  loads,  decreasing  to  a  slight 
extent  only  as  the  load  is  reduced.  In  the  other  sense,  suggested  in  Art.  342,  i.e. 
the  mechanical  efficiency  being  the  ratio  of  work  at  the  brake  to  gross  indicated 
work  (no  deduction  being  made  for  the  negative  loop  area  of  Fig.  136),  its  value 
falls  oif  sharply  as  the  load  decreases,  on  account  of  the  increased  proportion  of 
"fluid  friction."  Lucke  (39)  gives  the  following  as  average  values  for  the 
mechanical  efficiency  in  the  latter  sense  :  — 


ENGINE 

MECHANICAL 

EFFICIENCY 

four-cycle 

Two-cycle 

Large,  500  Ihp.  and  over,          .         .        . 

0.81  to  0.86 

0.63  to  0.70 

Medium,  25  to  500  Ihp.,            
Small,  4  to  25  Ihp.,           

0.79  to  0.81 
0.74  to  0.80 

0.64  to  0.66 
0.63  to  0.70 

346.  Heat  Balance.  The  principal  losses  in  the  gas  engine  are  due  to 
the  cooling  action  of  the  jacket  water  (a  necessary  evil  in  present  practice) 
and  to  the  heat  carried  away  in  the  exhaust.  The  arithmetical  means  of 
nine  trials  collated  by  the  writer  give  the  following  percentages  represent- 
ing the  disposition  of  the  total  heat  supplied:  to  the  jacket,  40.52;  to 
the  exhaust,  33.15;  work,  21.87;  unaccounted  for,  6.23.  Hutton  (40) 
tabulates  a  large  number  of  trials,  from  which  similar  arithmetical  aver- 
ages are  derived  as  follows  :  to  the  jacket,  37.96  ;  to  the  exhaust,  29.84  ; 
work,  22.24  ;  unaccounted  for,  8.6.  In  general,  the  larger  engines  show  a 
greater  proportion  of  heat  converted  to  work,  an  increased  loss  to  the 
exhaust,  and  a  decreased  loss  to  the  jacket. 


347.   Entropy  Diagram.     When  the  P  V  diagram  is  given,  points  may  be 

I^r  T) 

ferred  to  the  entropy  plane  by  the  formula  nb  -  na  =  fclog*~+  H°B*JT   (Art- 

V  a  *   a 

169).  The  state  a  may  be  taken  at  32°  F.  and  atmospheric  pressure  ;  then  the 
entropy  at  any  other  state  b  depends  simply  upon  Vb  and  Pb.  To  find  T7«,  we 
must  know  the  equation  of  the  gas.  According  to  Richmond  (41),  the  mean 


188 


APPLIED  THERMODYNAMICS 


value  of  k  may  be  taken  at  0.246  on  the  compression  curve  and  at  0.26  on  the  ex- 
pansion curve,  while  the  mean  values  of  I  corresponding  are  0.176  and  0.189.  The 
values  of  R  are  then  778(0.246  -  0.176)  =  54.46  and  778(0.260  -  0.189)  =  55.24. 
The  characteristic  equations  are,  then,  PV  —  54.46  T  along  the  compression  curve; 
and  PV  —  55.24  T  along  the  expansion  curve.  The  formula  gives  changes  of  en- 
tropy per  pound  of  substance.  The  indicator  diagram  does  not  ordinarily  depict 
the  behavior  of  one  pound;  but  if  the  weight  of  substance  used  per  cycle  be 
known,  the  volumes  taken  from  the  PV  diagram  may  be  converted  to  specific 
volumes  for  substitution  in  the  formula. 

It  is  sometimes. desirable  to  study  the  TV  relations  throughout  the  cycle.  In 
Fig.  154,  let  ABCD  be  the  PV  diagram.  Let  EF  be  any  line  of  constant  volume 
intersecting  this  diagram  at  G,  H.  By  Charles'  law,  TG :  TH : :  P0 :  PH.  The 


PdRT 


FIG.  154.     Art.  347.  —Gas  Engine  T V Diagram. 

ordinates  JG,  JH  may  therefore  serve  to  represent  temperatures  as  well  as  pres- 
sures, to  some  scale  as  yet  undetermined.  If  the  ordinate  JG  represent  tempera- 
ture, then  the  line  OG  is  a  line  of  constant  pressure.  Let  the  pressure  along  this 
line  on  a  TV  diagram  be  the  same  as  that  along  IG  on  a  PV  diagram.  Then 
(again  by  Charles'  law)  the  line  077  is  a  line  of  constant  pressure  on  the  TV  plane, 
corresponding  to  the  line  A'77  on  the  PV  plane.  Similarly,  OL  corresponds  to 
MN  and  OQ,  to  RB.  Project  the  points  S,  7',  7?,  73,  where  MN  and  7273  intersect 
the  PV  diagram,  until  they  intersect  OL,  OQ.  Then  points  U,  Q,  W,  X  are 
points  on  the  corresponding  TV  diagram.  The  scale  of  T  is  determined  from 
the  characteristic  equation;  the  value  of  R  may  be  taken  at  a  mean  between 
the  two  given.  A  transfer  may  now  be  made  to  the  NT  plane  by  the  aid  of  the 

equation  nb  -  na  -  I  loge  -£+(£_  /)iOg,  £t    (Art.    169),    in   which    Ta  =  491.6, 

7  a  Va 

54.46  x  491. ( 


2116.8 


=  12.64. 


GOVERNING 


189 


Figure  155,  from  Reeve  (42),  is  from  a  similar  four-cycle  engine.  The  enor- 
mous area  BA  CD  represents  heat  lost  to  the  water  jacket.  The  inner  dead  center 
of  the  engine  is  at  x ;  thereafter,  for  a  short 
period,  heat  is  evidently  abstracted  from  the 
fluid,  being  afterward  restored,  just  as  in  the 
case  of  a  steam  engine  (Art.  431),  because 
during  expansion  the  temperature  of  the  gases 
falls  below  that  of  the  cylinder  walls.  Reeve 
gives  several  instances  in  which  the  expansive 
path  resembles  xBzD ;  other  investigators  find 
a  constant  loss  of  heat  during  expansion.  Fig- 
ure 156  gives  the  PV  and  NT  diagrams  for 
a  Hornsby-Akroyd  engine;  the  expansion 
line  be  here  actually  rises  above  the  isothermal, 
indicative  of  excessive  after  burning. 

348.  Methods  of  Governing.  The  power 
exerted  by  an  Otto  cycle  engine  may 
be  varied  in  accordance  with  the  external 
load  by  various  methods;  in  order  that 
efficiency  may  be  maintained,  the  govern- 
ing should  not  lower  the  ratio  of  pressures  during  compression.  To  ensure 
this,  variation  of  the  clearance,  by  mechanical  means  or  water  pockets  and 
outside  compression  have  been  proposed,  but  no  practicably  efficient  means 

T 


FIG.  155.    Art.  347.  —  Gas  Engine 
Entropy  Diagram. 


V 


-N 


FIG.  156.    Art.  347.  —  Diagrams  for  Hornsby-Akroyd  Engine. 


have  yet  been  developed.  Automobile  engines  are  often  governed  by 
varying  the  point  of  ignition,  a  most  wasteful  method,  because  the  reduc- 
tion in  power  thus  effected  is  unaccompanied  by  any  change  whatever  in 
fuel  consumption.  Equally  wasteful  is  the  use  of  excessively  small  ports 
for  inlet  or  exhaust,  causing  an  increased  negative  loop  area  and  a  conse- 
quent reduction  in  power  when  the  speed  tends  to  increase.  In  engines 


190  APPLIED  THERMODYNAMICS 

where  the  combustion  is  gradual,  as  in  the  Brayton  or  Diesel,  the  point  of 
cut-off  of  the  charge  may  be  changed,  giving  the  same  sort  of  control  as  in 
a  steam  engine. 

Three  methods  of  governing  Otto  cycle  engines  are  in  more  or  less 
common  use.  In  the  " hitor-miss"  plan,  the  engine  omits  drawing  in  its 
charge  as  the  external  load  decreases.  One  or  more  idle  strokes  ensue. 
No  loss  of  economy  results  (at  least  from  a  theoretical  standpoint),  but  the 
speed  of  the  engine  is  apt  to  vary  on  account  of  the  increased  irregularity 
of  the  already  occasional  impulses.  Governing  by  changing  the  proportions 
of  the  mixture  (the  total  amount  being  kept  constant)  should  apparently 
not  affect  the  compression;  actually,  however,  the  compression  must  be 

fixed  at  a  sufficiently  low  point  to 
avoid  danger  of  pre-ignition  to  the 
strongest  probable  mixture,  and 
thus  at  other  proportions  the  de- 
gree of  compression  will  be  less 
than  that  of  highest  efficiency.  A 
change"  in  the  quantity  of  the  mix- 
ture, without  change  in  its  propor- 
tions, by  throttling  the  suction  or 
by  entirely  closing  the  inlet  valve 

~*   toward     the    end    of    the    suction 
FIG.  157.     Art.  348.  —  Effect  of  Throttling. 

stroke,  results  in  a  decided  change 

of  compression  pressure,  the  superimposed  cards  being  similar  to  those 
shown  in  Fig.  157.  In  theory,  at  least,  the  range  of  compression  pressures 
would  not  be  affected;  but  the  variation  in  proportion  of  clearance  gas 
present  requires  injurious  limitations  of  final  compression  pressure,  just 
as  when  governing  is  effected  by  variations  in  mixture  strength. 

349.  Defects  in  Gas  Engine  Governing.     The  hit-or-miss  system  may  be 
regarded  as  entirely  inapplicable  to  large  engines.     The  other  practicable 
methods  sacrifice  the  efficiency.     Further  than  this,  the  governing  influ- 
ence is  exerted  during  the  suction  stroke,  one  full  revolution  (in  four- 
cycle engines)  previous  to  the  working  stroke,  which  should  be  made  equal 
in  effort  to  the  external  load.     If  the  load  changes  during  the  intervening 
revolution,  the  control  will  be  inadequate.     Gas  engines  tend  therefore  to 
irregularity  in  speed  and  low  efficiency  under  variable  or  light  loads.    The 
first  disadvantage  is  overcome  by  increasing  the  number  of  cylinders,  the 
weight  of  the  fly  wheel,  etc.,  all  of  which  entails  additional  cost.     The  sec- 
ond disadvantage  has  not  yet  been  overcome. 

/  % 

350.  Construction  Details.     The  irregular  impulses  characteristic  of  the  gas 
engine  and  the   high  initial   pressures  attained   require    excessively  heavy  and 


DETAILS  191 

strong  frames.  For  anything  like  good  regulation,  the  fly  wheels  must  also  be 
exceptionally  heavy.  For  small  engines,  the  bed  casting  is  usually  a  single  heavy 
piece.  The  type  of  frame  usually  employed  on  large  engines  is  illustrated  in  Fig. 
152.  It  is  in  contact  with  the  foundation  for  its  entire  length,  and  in  many  cases 
is  tied  together  by  rods  at  the  top  extending  from  cylinder  to  cylinder. 

Each  working  end  of  the  cylinder  of  a  four-cycle  engine  must  have  two  valves, 
—  one  for  admission  and  one  for  exhaust.  In  many  cases,  three  valves  are  used, 
the  air  and  gas  being  admitted  separately.  The  valves  are  of  the  plain  disk  or 
mushroom  type,  with  beveled  seats ;  in  large  engines,  they  are  sometimes  of  the 
double-beat  type,  shown  in  Fig.  153.  Sliding  valves  cannot  be  employed  at  the 
high  temperature  of  the  gas  cylinder.  Exhaust  opening  must  always  be  under 
positive  control ;  the  inlet  valves  may  be  automatic  if  the  speed  is  low,  but  are 
generally  mechanically  operated  on  large  engines.  In  horizontal  four-cycle 
engines,  a  cam  shaft  is  driven  from  an  eccentric  at  half  the  speed  of  the  engine. 
Cams  on  this  shaft  operate  each  of  the  controlling  valves  by  means  of  adjustable 
oscillating  levers,  a  supplementary  spring  being  employed  to  accelerate  the  closing 
of  the  valves.  In  order  that  air  or  gas  may  pass  at  constant  speed  through  the 
ports,  the  cam  curve  must  be  carefully  proportioned  with  reference  to  the  varia- 
tion in  conditions  in  the  cylinder  (43).  Hutton  (44)  advises  proportioning  of 
ports  such  that  the  mean  velocity  may  not  exceed  60  ft.  per  second  for  automatic 
inlet  valves,  90  ft.  for  mechanically  operated  valves,  and  75  ft.  for  exhaust  valves, 
on  small  engines. 

351.  Starting  Gear.     No  gas  engine  is  self-starting.     Small  engines  are  often 
started  by  turning  the  fly  wheel  by  hand,  or  by  the  aid  of  a  bar  or  gearing.     An 
auxiliary  hand  air  pump  may  also  be  employed  to  begin  the  movement.     A  small 
electric  jnotor  is  sometimes  used  to  drive  a  gear -faced  fly  wheel  with  which  the 
motor  pinion  meshes.    In  all  cases,  the  engine  starts  against  its  friction  load  only, 
and  it  is  usual  to  provide  a  method  of  keeping  the  exhaust  valve  open  during  part 
of  the  compression  stroke  so  as  to  decrease  the  resistance.     In  multiple-cylinder 
engines,  as  in  automobiles,  the  ignition  is  checked  just  prior  to  stopping.     A  com- 
pressed but  unexploded  charge  will  then  often  be  available  for  restarting.     In  the 
Clerk  engine,  a  supply  of  unexploded  mixture  was  taken  during  compression  from 
the  cylinder  to  a  strong  storage  tank,  from  which  it  could  be  subsequently  drawn. 
Gasoline  railway  motor  cars  are  often  started  by  means  of  a  smokeless  powder 
cartridge  exploded  in  the  cylinder.     Modern  large  engines  are  started  by  com- 
pressed air,  furnished  by  a  direct-driven  or  independent  pump,  and  stored  in  small 
tanks. 

352.  Jackets.     The  use  of  water-spray  injection  during  expansion  has  been 
abandoned,  and  air  cooling   is   practicable  only  in    small   sizes.     The   cylinder, 
piston,  piston  rod,  and  valves  must  usually  be  thoroughly  water-jacketed.     Posi- 
tive circulation  must  be  provided,  and  the  water  cannot  be  used  over  again  unless 
artificially  cooled.     At  a  heat  consumption  of  200  B.  t.  u.  per  minute  per  I  hp., 
with  a  40  per  cent  loss  to  the  jacket,  the  theoretical  consumption  of  water  heated 
from  80  to  160°  F.  is  exactly  1  Ib.  per  Ihp.  per  minute.     This  is  greater  than  the 
water  consumption  of  a  non-condensing  steam  plant,  but  much  less  than  that  of 


192  APPLIED  THERMODYNAMICS 

a  condensing  plant.  The  discharge  water  from  large  engines  is  usually  kept 
below  130°  F.  In  smaller  units,  it  may  leave  the  jackets  at  as  high  a  temperature 
as  160°  F. 

353.  Possibilities  of  Gas  Power.  The  gas  engine,  at  a  comparatively  early 
stage  in  its  development,  has  surpassed  the  best  steam  engines  in  thermal  effi- 
ciency. Mechanically,  it  is  less  perfect  than  the  latter ;  and  commercially  it  is 
regarded  as  handicapped  by  the  greater  reliability,  more  general  field  of  applica- 
tion, and  much  lower  cost  (excepting,  possibly,  in  the  largest  sizes  *)  of  the  steam 
engine.  The  use  of  producer  gas  for  power  eliminates  the  coal  smoke  nuisance ; 
the  stand-by  losses  of  producers  are  low ;  and  gas  may  be  stored,  in  small  quanti- 
ties at  least.  The  small  gas  engine  is  quite  economical  and  may  be  kept  so.  The 
small  steam  engine  is  usually  wasteful.  The  Otto  cycle  engine  regulates  badly,  a 
disadvantage  which  can  be  overcome  at  excessive  cost ;  it  is  not  self -starting ;  the 
cylinder  must  be  cooled.  Even  if  the  mechanical  necessity  for  jacketing  could  be 
overcome,  the  same  loss  would  be  experienced,  the  heat  being  then  carried  off  in 
the  exhaust.  The  ratio  of  expansion  is  too  low,  causing  excessive  waste  of  heat 
at  the  exhaust,  which,  however,  it  may  prove  possible  to  reclaim.  The  heat  in  the 
jacket  water  is  large  in  quantity  and  low  in  temperature,  so  that  the  prob- 
lem of  utilization  is  confronted  with  the  second  law  of  thermodynamics. 
Methods  of  reversing  have  not  yet  been  worked  out,  and  no  important  marine 
applications  of  gas  power  have  been  made,  although  small  producer  plants  have 
been  installed  for  ferryboat  service  with  clutch  reversal,  and  compressed  and 
stored  gas  has  been  used  for  driving  river  steamers  in  France,  England,  and 
Germany. 

The  proposed  combinations  of  steam  and  gas  plants,  the  gas  plant  to  take  the 
uniform  load  and  the  steam  units  to  care  for  fluctuations,  really  beg  the  whole 
question  of  comparative  desirability.  The  bad  ''characteristic"  curve  —  low  effi- 
ciency at  light  loads  and  absence  of  bona  Ji.de  overload  capacity  —  will  always  bar 
the  gas  engine  from  some  services,  even  where  the  storage  battery  is  used  as  an 
auxiliary.  Many  manufacturing  plants  must  have  steam  in  any  case  for  process 
work.  In  such,  it  will  be  difficult  for  the  gas  engine  to  gain  a  foothold.  For  the 
utilization  of  blast  furnace  waste,  even  aside  from  any  question  of  commercial 
power  distribution,  the  gas  engine  has  become  of  prime  economic  importance. 

(1)  Hutton,  The  Gas  Engine,  1908,  545;  Clerk,  Theory  of  the  Gas  Engine,  1903, 
75.  (2)  Hutton,  The  Gas  Engine,  1908,  158.  (3)  Clerk,  The  Gas  Engine,  1890, 
119-121.  (4)  Ibid.,  129.  (5)  Ibid.,  133.  (6)  Ibid.,  137.  (7)  Ibid.,  198.  (8)  En- 
gineering News,  October  14,  1906,  357.  (9)  Lucke  and  Woodward,  Tests  of  Alcohol 
Fuel,  1907.  (10)  Junge,  Power,  December,  1907.  (10  a)  For  a  fuller  exposition  of  the 
limits  of  producer  efficiency  with  either  steam  or  waste  gas  as  a  diluent,  see  the  author's 
paper,  Trans.  Am.  Inst.  Chem.  Engrs.,  Vol.  IT.  (11)  Trans.  A  8.  M.  E.,  XXVIII,  6, 
1052.  (12)  A  test  efficiency  of  0.657  was  obtained  by  Parker,  Holmes,  and  Campbell : 
United  States  Geological  Survey,  Professional  Paper  No.  48.  (13)  Ewing,  The  Steam 
Engine,  1906,  418.  (14)  Clerk,  The  Theory  of  the  Gas  Engine,  1903.  (15)  Theorie 
und  Construction  eines  rationellen  Warmemotors.  (16)  Zeuner,  Technical  TJiermody- 
namics  (Klein),  1907,  I,  439.  (17)  Trans.  A.  S.  M.  E.,  XXI,  275.  (18)  Ibid.,  286. 

*  Piston  speeds  of  large  gas  engines  may  exceed  those  of  steam  engines. 


GAS   POWER  193 

(19)  Op.  cit.,  XXIV,  171.  (20)  Op.  cit.,  XXI,  276.  (21)  Gas  Engine  Design,  1897, 
33.  (22)  Op.  cit.,  p.  34  et  seq.'  (23)  See  Lucke,  Trans.  A.  8.  M.  E.,  XXX,  4,  418. 
(24)  The  Gas  Engine,  1890,  p.  95  et  seq.  (25)  A.  L.  Westcott,  Some  Gas  Engine  Cal- 
culations based  on  Fuel  and  Exhaust  Gases :  Power,  April  13,  1909,  p.  693.  (26)  Hut- 
ton,  The  Gas  Engine,  1908,  pp.  507,  522.  (27)  The  Gas  Engine,  1908.  (28)  Clerk, 
op.  cit.,  p.  216.  (29)  Op.  cit.,  p.  291.  (30)  Op.  cit.,  p.  38.  The  corresponding  usual 
mean  effective  pressures  are  given  on  p.  36.  (30a)  See  the  author's  paper,  Commer- 
cial Ratings  for  Internal  Combustion  Engines,  in  Machinery,  April,  1910.  (31)  Zeits. 
Ver.  Deutsch.  Ing.,  November  24,  1906  ;  Power,  February,  1907.  (32)  The  Electrical 
World,  December  7,  1907,  p.  1132.  (33)  Trans.  A.  8.  M.  E.,  XXIV,  1065.  (34)  Bui. 
Soc.  de.V  Industrie  Mineral,  Ser.  Ill,  XIV,  1461.  (35)  Trans.  A.  S.  M.  E.,  XXVIII, 
6,  1041.  (36)  Quoted  by  Mathot,  supra.  (37)  Also  from  Mathot.  (38)  Mathot, 
supra.  (39)  Op.  cit.,  p.  5.  (40)  Op.  cit.,  pp.  342-343.  (41)  Trans.  A.  S.  M.  E.,  XIX, 
491.  (42)  Ibid.,  XXIV,  171.  (43)  Lucke,  Gas  Engine  Design,  1905.  (44)  Op.  cit., 
483. 

SYNOPSIS  OF   CHAPTER  XI 
The  Producer 

The  importance  of  the  gas  engine  is  largely  due  to  the  producer  process  for  making 

cheap  gas. 
In  the  gas  engine,  combustion  occurs  in  the  cylinder,  and  the  highest  temperature 

attained  by  the  substance  determines  the  cyclic  efficiency. 
Fuels  are  natural  gas,  carbureted  and  un carbureted  water  gas,  coal  gas,  coke  oven 

gas,    producer  gas,   blast  furnace  gas ;    gasoline,   kerosene,   fuel   oil,   distillate, 

alcohol,  coal  tars. 
The  gas  producer  is  a  lined  cylindrical  shell  in  which  the  fixed  carbon  is  converted 

into  carbon  monoxide,  while  the  hydrocarbons  are  distilled  off,  the  necessary  heat 

being  supplied  by  the  fixed  carbon  burning  to  CO. 
The  maximum  theoretical  efficiency  of  the  producer  making  power  gas  is  less  than  that 

of  the  steam  boiler.     Either  steam  or  exhaust  gas  from  the  engine  must  be  intro- 
duced to  attain  maximum  efficiency. 
The  mean  composition  of  producer  gas,  by  volume,  is  CO,  19.2 ;  CO2,  9.5  ;  H,  12.4  ; 

CH4,  C2H4,  3.1  ;  N,  55.8. 
The  "figure  of  merit"  is  the  heating  value  of  the  gas  per  pound  of  carbon  contained. 


Gas  Engine  Cycles 

The  Otto  cycle  is  bounded  by  two  adiabatics  and  two  lines  of  constant  volume ;  the 

engine  may  operate  in  either  the  four-stroke  cycle  or  the  two-stroke  cycle. 
In  the  two-stroke  cycle,  the  inlet  and  exhaust  ports  are  both  open  at  once. 

In  the  Otto  cycle,  — 5  =  — '  and  ~6  =  ^- 
Pe      Pd          Te      Td 

Efficiency  =  Te  ~  Td  =  1  —  (— * VV  =  Th  ~  Tf=  1  —  i-f\  ~~y~;  it  depends  solely  on  the 
Te  \  Pe  I  Tb  \  1  b  I 

extent  of  compression. 

Efficiency  of  Atkinson  engine  (isothermal  rejection  of  heat)  =  1  —       _        log*  — ; 
higher  than  that  of  the  Otto  cycle. 


194  APPLIED  THERMODYNAMICS 

rrj  rrt  rwi     ^^     rrt 

Lenoir  cycle:  constant  pressure  rejection  of  heat;  efficiency  •=!— _JLZ — *_y_J * 

If  -ld  if—   id 

Brayton  cycle  :  combustion  at  constant  pressure :  efficiency  =  1 g~~    * — * — -£  ; 

y(lb—ln)      lb—  Ln 

fJJ  fJ-9 

or,  with  complete  expansion,     n.~ — -> 

Tn 

A  special  comparison  shows  the  Clerk  Otto  engine  to  give  a  much  higher  efficiency  than 
the  Brayton  or  Lenoir  engine,  but  that  the  Brayton  engine  gives  slightly  the  largest 
work  area. 

The  Cleric  Otto  (complete  pressure)  cycle  gives  an  efficiency  of  1—  ; f~  r*— y    9  ~      , 

jT«  —  Tc       Te  —  Tc 

intermediate  between  that  of  the  ordinary  Otto  and  the  Atkinson. 


The  Diesel  cycle:  isothermal  combustion;  efficiency  =1 = — ;  increases 

as  ratio  of  expansion  decreases.  yHTa  loge  — 


Modifications  in  Practice 

The  PV  diagram  of  an  actual  Otto  cycle  engine  is  influenced  by 

(a)  proportions  of  the  mixture,  which  must  not  be  too  weak  or  too  strong,  and 

must  be  controllable ; 

(6)  maximum  allowable  temperature  after  compression  to  avoid  pre-ignition ;  the 
range  of  compression,  which  determines  the  efficiency,  depends  upon  this  as 
well  as  upon  the  pre-compression  pressure  and  temperature  ; 

(c)  the  rise  of  pressure  and  temperature  during  combustion ;  always  less  than 

those  theoretically  computed,  on  account  of  (1)  divergences  from  Charles' 
law,  (2)  the  variable  specific  heats  of  gases,  (3)  slow  combustion,  (4)  disso- 
ciation ; 

(d)  the  shape  of  the  expansion  curve,  usually  above  the  adiabatic,  on  account  of 

after  burning,  in  spite  of  loss  of  heat  to  the  cylinder  wall; 

(e)  the  forms  of  the  suction  and  exhaust  lines,  which  may  be  affected  by  badly 

proportioned  ports  and  passages  and  by  improper  valve  action. 
Dissociation  prevents  the  combustion  reaction  of  more  than  a  certain  proportion  of 

the  elementary  gases  at  each  temperature  within  the  critical  limits. 
The  j)oi'n£  of  ignition  must  somewhat  precede  the  end  of  the  stroke,  particularly  with 

weak  mixtures. 

Methods  of  ignition  are  by  hot  tube,  jump  spark,  and  make  and  break. 
Cylinder  clearance  ranges  from  8.7  to  56  per  cent.     It  is  determined  by  the  compression 

pressure  range. 
Scavenging  is  the  expulsion  of  the  burnt  gases  in  the  clearance  space  prior  to  the 

suction  stroke. 
The  diagram  factor  is  the  ratio  of  the  area  of  the  indicator  diagram  to  that  of  the  ideal 

cycle. 

Mean  effective  pressure  =• 


Vd-Ve 


GAS  POWER  195 

Gas  Engine  Design 

In  designing  an  engine  for  a  given  power,  the  gas  composition,  rotative 
speed  and  piston  speed  are  assumed.  The  probable  efficiency  may  be 
estimated  in  advance.  Overload  capacity  must  be  secured  by  assum- 
ing a  higher  capacity  than  that  normally  needed ;  the  engine  will  do 
no  more  work  than  that  for  which  it  is  designed. 

Current  Forms 

Otto  cycle  oil  engines  include  the  Mietz  and  Weiss,  two-cycle,  and  the  Daimler,  Priest- 
man,  and  Hornsby-Akroyd,  four-cycle. 

Modern  forms  of  the  Otto  gas  engine  include  the  Otto,  Foos,  Crossley-Otto,  and 
Andrews. 

The  Westinghouse,  Riverside,  and  Allis-Chalmers  engines  are  built  in  the  largest  sizes. 

Two-cycle  gas  engines  include  the  Oechelhaueser  and  Koerting. 

Special  engines  are  built  for  motor  bicycles,  automobiles,  and  launches,  and  for  burn- 
ing alcohol. 

The  basis  of  efficiency  is  the  heat  unit  consumption  per  horse  power  per  minute. 

The  mechanical  efficiency  may  be  computed  from  either  gross  or  net  indicated  work. 

Recorded  efficiencies  of  gas  engines  range  up  to  42.7 per  cent;  plant  efficiencies  to  0.7 
Ib.  coal  per  brake  hp.-hr. 

The  mechanical  efficiency  increases  with  the  size  of  the  engine,  and  is  greater  with  the 
four-stroke  cycle. 

About  38  per  cent  of  the  heat  supplied  is  carried  off  by  the  jacket  water,  and  about 
S3  per  cent  by  the  exhaust  gases,  in  ordinary  practice. 

The  entropy  diagram  may  be  constructed  by  transfer  from  the  PFor  TV  diagrams. 

Governing  is  effected 

(#)  by  the  hit-or-miss  method;  economical,  but  unsatisfactory  for  speed  regulation, 

(6)  by  throttling,  }  , 

\  both  wasteful, 
(c)  by  changing  mixture  proportions,  J 

In  all  cases,  the  governing  effort  is  exerted  too  early  in  the  cycle. 

Gas  engines  must  have  heavy  frames  and  fly  wheels ;  exhaust  valves  (and  inlet  valves 
at  high  speed)  must  be  mechanically  operated  by  carefully  designed  cams;  pro- 
vision must  be  made  for  starting  ;  cylinders  and  other  exposed  parts  are  jacketed. 
About  1  Ib.  of  jacket  water  is  required  per  Ihp. -minute. 

Gas  engine  advantages:  high  thermal  efficiency  ;  elimination  of  coal  smoke  nuisance  ; 
stand-by  losses  are  low ;  gas  may  be  stored  ;  economical  in  small  units ;  desirable 
for  utilizing  blast  furnace  gas. 

Disadvantages :  mechanically  still  evolving  ;  of  unproven  reliability  ;  less  general  field 
of  application  ;  generally  higher  first  cost ;  poor  regulation  ;  not  self-starting ; 
cylinder  must  be  cooled  ;  low  ratio  of  expansion ;  non-reversible ;  no  overload 
capacity  ;  no  available  by-product  heat  for  process  work  in  manufacturing  plants. 

PROBLEMS 

1.    Compute  the  volume  of  air  ideally  necessary  for  the  complete  combustion  of 
1  cu.  ft.  of  gasoline  vapor,  C6H14. 


196  APPLIED  THERMODYNAMICS 

2.  Find  the  maximum  theoretical  efficiency,  using  pure  air  only,  of  a  power  gas 
producer  fed  with  a  fuel  consisting  of  70  per  cent  of  fixed  carbon  and  30  per  cent  of 
volatile  hydrocarbons. 

3.  In  Problem  2,  what  is  the  theoretical  efficiency  if  20  per  cent  of  the  oxygen 
necessary  for  gasifying  the  fixed  carbon  is  furnished  by  steam  ? 

4.  In  Problem  3,  if  the  hydrocarbons  (assumed  to  pass  off  unchanged)  are  half 
pure  hydrogen  and  half  marsh  gas,  compute  the  producer  gas  composition  by  volume, 
using  specific  volumes  as  follows :    nitrogen,  12.75  ;  hydrogen,  178.83 ;  carbon  mo- 
noxide, 12.75  ;  marsh  gas,  22.3. 

5.  A  producer  gasifying  pure  carbon  is  supplied  with  the  theoretically  necessary 
amount  of  oxygen  from  the  atmosphere  and  from  the  gas  engine  exhaust.     The  latter 
consists  of  28.4  per  cent  of  CO2  and  71.6  per  cent  of  N,  by  weight,  and  is  admitted  to 
the  extent  of  1  Ib.  per  pound  of  pure  carbon  gasified.     Find  the  rise  in  temperature, 
the  composition  of  the  produced  gas,  and  the  efficiency  of  the  process.     The  heat  of 
decomposition  of  C02  to  C  may  be  taken  at  14,500  B.  t.  u.  per  pound  of  carbon. 

6.  Find  the  figures  of  merit  in  Problem's  4  and  5.     (Take  the  heating  value  of  H 
at  53,400  ;  of  CH4,  at  22,500.) 

7.  In  Fig.  134,  let  -^  =  4,  Pd  =  30,  Pg  =  Pgn  =  Pd  +  10,  Th  =  3000°,  Td  =  1000° 

Ve 

(absolute).    Find  the  efficiency  and  area  of  each  of  the  ten  cycles,  for  1  Ib.  of  air,  with- 
out using  efficiency  formulas. 

8.  In  Problem  7,  show  graphically  by  the  NT  diagram  that  the  Carnot  cycle  is 
the  most  efficient. 

9.  What  is  the  maximum  theoretical  efficiency  of  an  Otto  four-cycle  engine  in 
which  the  fuel  used  is  producer  gas?     (See  Art.  312.) 

10.  What  maximum  temperature  should  theoretically  be  attained  in  an  Otto  en- 
gine using  gasoline,  with  a  temperature  after  compression  of  780°  F.  ?    (The  heat  liber- 
ated by  the  gasoline,  available  for  increasing  the  temperature,  may  be  taken  at  19,000 
B.  t.  u.  per  pound.) 

11.  Find  the  mean  effective  pressure  and  the  work  done  in  an  Otto  cycle  between 
volume  limits  of  0.5  and  2.0  cu.  ft.  and  pressure  limits  of  14.7  and  200  Ib.  per  square 
inch  absolute. 

12.  An  Otto  engine  is  supplied  with  pure  CO,  with  pure  air  in  just  the  theoretical 
amount  for  perfect  combustion.    Assume  that  the  dissociation  effect  is  indicated  by  the 
formula*  (1.00  —  a) (6000  —  T)=300,  in  which  a  is  the  proportion  of  gas  that  will 
combine  at  the  temperature   T°  F.    If  the  temperature  after  compression  is  800°  F., 
what  is  the  maximum  temperature  attained  during  combustion,  and  what  proportion 
of  the  gas  will  burn  during  expansion  and  exhaust,  if  the  combustion  line  is  one  of  con- 
stant volume  ?    The  value  of  I  for  CO  is  0.1758. 

13.  An  Otto  engine  has  a  stroke  of  24  in.,  a  connecting  rod  60  in.  long,  and  a  pis- 
ton speed  of  400  ft.  per  minute.     The  clearance  is  20  per  cent  of  the  piston  displace- 
ment, and  the  volume  of  the  gas,  on  account  of  the  speed  of  the  piston  as  compared 
with  that  of  the  flame,  is  doubled  during  ignition.    Plot  its  path  on  the  P  V  diagram 

*  This  is  assumed  merely  for  illustrative  purposes.    It  has  no  foundation  and  is  irra- 
tional at  limiting  values. 


GAS  POWER  197 

and  plot  the  modified  path  when  the  piston  speed  is  increased  to  800  ft.  per  minute, 
assuming  the  flame  to  travel  at  uniform  speed  and  the  pressure  to  increase  directly  as 
the  spread  of  the  flame.  The  pressure  range  during  ignition  is  from  100  to  200  Ib. 

14.  The  engine  in  Problem  11  is  four-cycle,  two-cylinder,  double-acting,  and  makes 
100  r.  p.  m.  with  a  diagram  factor  of  0.40.     Find  its  capacity. 

15.  Starting  at  Pd  =  14.7,  Vd  =  43.45,  Td  =  32°  F.  (Fig.  122),  plot  (a)  the  ideal 
Otto  cycle  for  1  lb.  of  CO  with  the  necessary  air,  and  (&)  the  probable  actual  cycle 
modified  as  described  in  Arts.  309-328,  and  find  the  diagram  factor.     Clearance  is  25 
per  cent  of  the  piston  displacement  in  both  cases. 

16.  Find  the  cylinder  dimensions  in  Art.  332  if  the  gas  composition  be  as  given  in 
Art.  285.    (Take  the  average  heating  value  of  CII4  and  C2H4  at  22,500  B.  t.  u.  per  pound, 
and  assume  that  the  gas  contains  the  same  amount  of  each  of  these  constituents.) 

17.  Find  the  clearance,  cylinder  dimensions,  and  probable  efficiency  in  Art.  332  if 
the  engine  is  two-cycle. 

18.  Find  the  size  of  cylinders  of  a  four-cylinder,  four-cycle,  single-acting  gasoline 
engine  to  develop  30  bhp.  at  1200  r.  p.  m.,  the  cylinder  diameter  being  equal  to  the 
stroke.    Estimate   its  thermal  efficiency,  the  theoretically  necessary  quantity  of  air 
being  supplied. 

19.  An  automobile  consumes  1  gal.  of  gasoline  per  9  miles  run  at  50  miles  per 
hour,  the  horse  power  developed  being  25.     Find  the  heat  unit  consumption  per  Ihp. 
per  minute  and  the  thermal  efficiency  ;  assuming  gasoline  to  weigh  7  lb.  per  gallon. 

20.  A  two-cycle  engine  gives  an  indicator  diagram  in  which  the  positive  work 
area  is  1000  ft.-lb.,  the  negative  work  area  90  ft.-lb.    The  work  at  the  brake  is  700 
ft.-lb.     Give  two  values  for  the  mechanical  efficiency. 

21.  The  engine  in  Problem  17  discharges  30  per  cent  of  the  heat  it  receives  to  the 
jacket.     Find  the  water  consumption  in  pounds  per  minute,  if  its  initial  temperature 
is  72°  F. 

22.  In  Art.  344,  what  was  the  producer  efficiency  in  the  case  of  the  Guldner  en- 
gine, assuming  its  mechanical  efficiency  to  have  been  0.85  ?    If  the  coal  contained 
13,800  B.  t.  u.  per  pound,  what  was  the  coal  consumption  per  brake  hp.-hr.  ? 

23.  Given  the  indicator  diagram  of  Fig.  158,  plot  accurately  the  TV  diagram,  the 
engine  using  0.0462  lb.  of  substance  per  cycle.    Draw  the  compressive  path  on  the  NT 
diagram  by  both  of  the  methods  of  Art.  347. 

24.  The  engine  in  Problem  17  governs  by  throttling  its  charge.     To  what  percent- 
age of  the  piston  displacement  should  the  clearance  be  decreased  in  order  that  the  pres- 
sure after  compression  may  be  unchanged  when  the  pre-compression  pressure  drops  to 
10  lb.  absolute  ?     What  would  be  the  object  of  such  a  change  in  clearance  ? 

25.  In  the  Diesel  engine,  Problem  7,  by  what  percentages  will  the  efficiency  and 
capacity  be  affected,  theoretically,  if  the  supply  of  fuel,  is  cut  off  50  per  cent  earlier  in 

the  stroke  ?     (i.e.,  cut-off  occurs  when  the  volume  is      ft  ~     "  +  Fa,  Fig.  134.) 

26.  For  Martin's  project  (Art.  276),  determine  the  velocity  of  the  gas  in  the  pipe 
line  if  it  is  transmitted  300  miles.     Confirm,  approximately,  the  estimate  of  power  con- 
sumption, the  plant  operating  continuously.     If  the  coal  contains  12,000  B.  t.  u.  per 
pound  and  cost  50  cents  per  2000  lb.,  and  the  gas  contains  600  B.  t.  u.  per  cubic  foot, 
what  is  the  efficiency  of  the  coal  gas  retorts  ? 


198 


APPLIED  THERMODYNAMICS 


300 


240 


120 


60 


0.20  040  0.60  0.80  1.00 

FIG.  158.    Prob.  23.— Indicator  Diagram  for  Transfer. 


CUBIC  FEET 


27.  Under  the  conditions  of  Art.  335,  develop  a  relation  between  piston  displace- 
ment in  cubic  inches  per  minute,  and  Ihp.,  for  four  cylinder  four-cycle  single  acting 
gasolene  engines.  Also  find  the  relation  between  cylinder  volume  and  Ihp.  if  engines 
run  at  1500  r.  p.  m.,  and  the  relation  between  cylinder  diameter  and  Ihp.  if  bore  =  stroke, 
at  1500  r.  p.  m. 

2£L  In  an  Otto  engine,  the  range  of  pressures  during  compression  is  from  13  to 
130  lb.,  the  compression  curve  pv1'8  =  c.  Find  the  percentage  of  clearance. 


CHAPTER   XII 
THEORY  OF  VAPORS 

354.  Boiling  of  Water.     If  we  apply  heat  to  a  vessel  of  water  open 
to  the  atmosphere,  an  increase  of  temperature  and  a  slight  increase 
of  volume  may  be  observed.     The  increase  of  temperature  is  a  gain 
of  internal  energy;  the  slight  increase  of  volume  against  the  constant 
resisting  pressure  of  the  atmosphere  represents  the  performance  of 
external  work,  the  amount  of  which  may  be  readily  computed.     After 
this  operation  has  continued  for  some  time,  a  temperature  of  212°  F. 
is  attained,  arid  steam  begins  to  form.     The  water  now  gradually 
disappears.     The  steam  occupies  a  much  larger  space  than  the  water 
from  which  it  was  formed  ;  a  considerable  amount  of  external  work  is 
done  in  thus  augmenting  the  volume  against  atmospheric  pressure ; 
and  the  common  temperature  of  the  steam  and  the  water  remains  con- 
stant at  212°  F.  during  evaporation. 

355.  Evaporation  under  Pressure.     The  same  operation  may  be 
performed  in  a  closed  vessel,  in  which  a  pressure  either  greater  or  less 
than  that  of  the  atmosphere  may  be  maintained.     The  water  will  now 
boil  at  some  other  temperature  than  212°  F. ;  at  a  lower  temperature, 
if  the  pressure  is  less  than  atmospheric,  and  at  a  higher  temperature,  if 
greater.     The  latter  is  the  condition  in  an  ordinary  steam  boiler.     If 
the  water  be  heated  until  it  is  all  boiled  into  steam,  it  will  then  be 
possible  to  indefinitely  increase  the  temperature  of  the  steam,  a  result 
not  possible  as  long  as  any  liquid  is  present.     The  temperature  at 
which    boiling   occurs  may   range   from  32°  F.  for   a   pressure   of 
0.089   Ib.    per   square    inch,    absolute,   to   428°   F.   for   a   pressure 
of  336  Ib. ;    but  for  each  pressure  there  is  a  fixed  temperature  of 
ebullition. 

356.  Saturated  Vapor.     Any  vapor  in  contact  with  its  liquid  and 
in  thermal  equilibrium  (i.e.  not  constrained  to  receive  or  reject  heat) 

199 


200  APPLIED  THERMODYNAMICS 

is  called  a  saturated  vapor.  It  is  at  the  minimum  temperature  (that 
of  the  liquid)  which  is  possible  at  the  existing  pressure.  Its  density 
is  consequently  the  maximum  possible  at  that  pressure.  Should  it 
be  deprived  of  heat,  it  cannot  fall  in  temperature  until  after  it  has 
been  first  completely  liquefied.  If  its  pressure  is  fixed,  its  temperature 
and  density  are  also  fixed.  Saturated  vapor  is  then  briefly  definable 
as  vapor  at  the  minimum  temperature  or  maximum  density  possible 
under  the  imposed  pressure. 

357.  Superheated  Vapor.  A  saturated  vapor  subjected  to  ad- 
ditional heat  at  constant  pressure,  if  in  the  presence  of  its  liquid, 
cannot  rise  in  temperature ;  the  only  result  is  that  more  of  the  liquid 
is  evaporated.  When  all  of  the  liquid  has  been  evaporated,  or  if  the 
vapor  is  conducted  to  a  separate  vessel  where  it  may  be  heated  while 
not  in  contact  with  the  liquid,  its  temperature  may  be  made  to  rise, 
and  it  becomes  a  superheated  vapor.  It  may  be  now  regarded  as  an 
imperfect  gas;  as  its  temperature  increases,  it  constantly  becomes 
more  nearly  perfect.  Its  temperature  is  always  greater,  and  its 
density  less,  than  those  properties  of  saturated  vapor  at  the  same 
pressure ;  either  temperature  or  density  may,  however,  be  varied  at 
will,  excluding  this  limit,  the  pressure  remaining  constant.  At 
constant  pressure,  the  temperature  of  steam  separated  from  water 
increases  as  heat  is  supplied. 

The  characteristic  equation,  PV  =  RT,  of  a  perfect  gas  is  inapplicable  to  steam. 
(See  Art.  390.)  The  relation  of  pressure,  volume,  and  temperature  is  given  by 
various  empirical  formulas,  including  those  of  Joule  (1),  Rankine  (2),  Him  (3), 
Racknel  (4),  Clausius  (5),  Zeuner  (6),  and  Knoblauch  Linde  and  Jakob  (7). 
These  are  in  some  cases  applicable  to  either  saturated  or  superheated  steam. 

SATURATED  STEAM 

358.  Thermodynamics  of  Vapors.  The  remainder  of  this  text  is 
chiefly  concerned  with  the  phenomena  of  vapors  and  their  application 
in  vapor  engines  and  refrigerating  machines.  The  behavior  of  vapors 
during  heat  changes  is  more  complex  than  that  of  perfect  gases. 
The  temperature  of  boiling  is  different  for  different  vapors,  even  at 
the  same  pressure;  but  the  following  laws  hold  for  all  other  vapors 
as  well  as  for  that  of  water : 


FORMATION   OF  STEAM  201 

(1)  The  temperatures  of  the  liquid  and  of  the  vapor  in  contact  with 

it  are  the  same  ; 

(2)  The  temperature  of  a  specific  saturated  vapor  at  a  specified  pres- 

sure is  always  the  same ; 

(3)  The  temperature  and  the  density  of  a  vapor  remain  constant 

during  its  formation  from  liquid  at  constant  pressure ; 

(4)  Increase  of  pressure  increases  the  temperature  and  the  density  of 

the  vapor ;  * 

(5)  Decrease  of  pressure  lowers  the  temperature  and  the  density ; 

(6)  The  temperature  can  be  increased  and  the  density  can  be  decreased 

at  will,  at  constant  pressure,  when  the  vapor  is  not  in  contact 
with  its  liquid ; 

(7)  If  the  pressure  upon  a  saturated  vapor  be  increased  without  allow- 

ing its  temperature  to  rise,  the  vapor  must  condense  ;  it  cannot 
exist  at  the  increased  pressure  as  vapor  (Art.  356).  If  the 
pressure  is  lowered  while  the  temperature  remains  constant,  the 
vapor  becomes  superheated. 

359.  Effects  of  Heat  in  the  Formation  of  Steam.  Starting  with 
a  pound  of  water  at  32°  F.,  as  a  convenient  reference  point,  the  heat 
expended  during  the  formation  of  saturated  steam  at  any  temperature 
and  pressure  is  utilized  in  the  following  ways : 

(1)  h  units  in  the  elevation  of  the  temperature  of  the  water.  If  the 
specific  heat  of  water  be  unity,  and  t  be  the  boiling  point, 
h  =  t  —  32 ;  actually,  h  always  slightly  exceeds  this,  but  the 
excess  is  ordinarily  small,  f  J 

*  Since  mercury  boils,  at  atmospheric  pressure,  at  675°  F.,  common  thermometers 
cannot  be  used  for  measuring  temperatures  higher  than  this  ;  but  by  filling  the  space  in 
the  thermoinetric  tube  above  the  mercury  with  gas  at  high  pressure,  the  boiling  point 
of  the  mercury  may  be  so  elevated  as  to  permit  of  its  use  for  measuring  flue  gas 
temperatures  exceeding  800°  F. 

t  According  to  Barnes'  experiments  (8),  the  specific  heat  of  water  decreases  from 
1.0094  at  32°  F.  to  0.90735  at  100°  F..  and  then  steadily  increases  to  1.0476  at  428°  F. 

t  In  precise  physical  experimentation,  it  is  necessary  to  distinguish  between  the 
value  of  h  measured  above  32°  F.  and  atmospheric  pressure,  and  that  measured  above 
32°  F.  and  the  corresponding  pressure  of  the  saturated  vapor.  This  distinction  is  of  no 
consequence  in  ordinary  engineering  work. 


202  APPLIED  THERMODYNAMICS 

(2)  P{     ~~  v'  units  in  the  expansion  of  the  water  (external  work),  p 

778 

being  the  pressure  per  square  foot  and  v  and  F^the  initial  and 
final  specific  volumes  of  the  water  respectively.  This  quantity 
is  included  in  item  h,  as  above  defined ;  it  is  so  small  as  to  be 
usually  negligible,  and  the  total  heat  required  to  bring  the 
water  up  to  the  boiling  point  is  regarded  as  an  internal  energy 
change. 

(3)  e  =  ^ — — -  units  to  perform  the  external  work  of  increasing 

778 

the  volume  at  the  boiling  point  from  that  of  the  water  to  that  of 
the  steam,  TFbeing  the  specific  volume  of  the  steam. 

(4)  r  units  to  perform  the  disgregation  work  of  this  change  of  state 
(Art.  15)  ;  items  (3)  and  (4)  being  often  classed  together  as  L. 

The  total  heat  expended  per  pound  is  then 
H=  h+L=  h  +  r+e. 

The  values  of  these  quantities  vary  widely  with  different  vapors,  even  when 
at  the  same  temperature  and  pressure;  in  general,  as  the  pressure  increases,  h 
increases  and  L  decreases.  Watt  was  led  to  believe  (erroneously)  that  the  sum  of 
k  and  L  for  steam  was  a  constant;  a  result  once  described  as  expressing  "  Watt's 
Law."  This  sum  is  now  known  to  slowly  increase  with  increase  of  pressure. 

360.  Properties  of  Saturated  Steam.  It  has  been  found  experimentally 
that  as  p,  the  pressure,  increases,  t,  h,  e,  and  //  increase,  while  r  and  L 
decrease.  These  various  quantities  are  tabulated  in  what  is  known  as  a 
steam  table.*  A  convenient  form  of  table  for  quick  reference  is  that  in 

*  Regnault's  experiments  were  the  foundation  of  the  steam  tables  of  Rankine  (9), 
Zeuner  (10),  and  Porter  (11).  The  last  named  have  been  regarded  as  extremely  accurate, 
and  were  adopted  as  standard  for  use  in  reporting  trials  of  steam  boilers  and  pumping 
engines  by  the  American  Society  of  Mechanical  Engineers.  They  do  not  give  all  of  the 
thermal  properties,  however,  and  have  therefore  been  unsatisfactory  for  some  purposes. 
The  tables  of  D  welsh  auevers-Dery  (12)  were  based  on  Zeuner's;  Buel's  tables,  origi- 
nally published  in  Weisbach's  Mechanics  (13),  on  Rankine's.  The  table  in  Kent's 
Mechanical  Engineers'1  Pocket-Book  is  derived  from  Dwelshauevers-Dery  and  Buel. 
Peabody's  tables  are  computed  directly  from  Regnault's  work  (14).  The  principal 
differences  in  these  tables  were  due  to  some  uncertainty  as  to  the  specific  volume  of 
steam  (15).  The  precise  work  of  Holborn  and  Henning  (16)  on  the  pressure-tempera- 
ture relation  and  the  adaptation  by  Davis  (17)  of  recent  experiments  on  the  specific 
heat  of  superheated  steam  to  the  determination  of  the  total  heat  of  saturated  steam  (Art. 
388)  have  suggested  the  possibility  of  steam  tables  of  greater  accuracy.  The  most 
recent  and  satisfactory  of  these  is  that  of  Marks  and  Davis  (18),  values  from  which 
are  adopted  in  the  remainder  of  the  present  text.  (See  pp.  247,  248.) 


Temperature  Fahrenheit. 


204  APPLIED  THERMODYNAMICS 

which  the  values  are  plotted  as  coordinates,  as  in  Fig.  159.  Using  this, 
we  may  find  and  check  numerical  values  for  the  items  in  Art.  359,  remem- 
bering that  V=  0.017,  as  follows  : 

p  =  14.697  (Ib.  per  sq.  in.).  p  =  100.58  (Ib.  per  sq.  in.). 


h  =  180.6.  ft  =  298.4. 

H=  1146.6.  H=  1182.0. 

L  =  965.8.  L=  883.6. 

r  =  893.5.  r  =  802.4. 

e  =  72.3.  e  =  81.2. 

Our  original  knowledge  of  these  values  was  derived  from  the  compre- 
hensive experiments  of  Regnault,  whose  empirical  formula  for  the  total 
heat  of  saturated  steam  was  H=  1081.94  +  0.305  t.  The  recent  investi- 
gations of  Davis  (17)  show,  however,  that  a  more  accurate  expression  is 

H  =  1150.3  +  0.3745  (t  -  212)  -  0.00055  (t  -  212)2  (Art.  388). 

(The  total  heat  at  212°  F.  is  represented  by  the  value  1150.3.)  Barnes' 
and  other  determinations  of  the  specific  heat  of  water  permit  of  the  com- 
putation of  h  ;  and  L  —  H—li.  The  value  of  e  may  be  directly  calculated 
if  the  volume  W  is  known,  and  r  =  L  —  e.  An  inspection  of  Fig.  159  shows 
that  the  value  of  r  has  a  straight  line  relation,  approximately,  with  the 
temperature.  This  may  be  expressed  by  the  formula  r  =  1061.3  —  0.79  t°  F. 
The  method  of  deriving  the  steam  volume,  always  tabulated  with  these 
other  thermal  properties,  will  be  considered  later.  When  saturated  steam 
is  condensed,  all  of  the  heat  quantities  mentioned  are  emitted  in  the 
reverse  order,  so  to  speak.  Regnault's  experiments  were  in  fact  made, 
not  by  measuring  the  heat  absorbed  during  evaporation,  but  that  emitted 
during  condensation.  Items  h  and  r  are  both  internal  energy  effects  ; 
they  are  sometimes  grouped  together  and  indicated  by  the  symbol  E-, 
whence  H=E  +  e.  The  change  of  a  liquid  to  its  vapor  furnishes  the 
best  possible  example  of  what  is  meant  by  disgregation  work.  If  there  is 
any  difficulty  in  conceiving  what  such  work  is,  one  has  but  to  compare  the 
numerical  values  of  L  and  r  for  a  given  pressure.  What  becomes  of  the 
difference  between  L  and  e?  The  quantity  L  is  often  called  the  latent 
heat,  or,  more  correctly,  the  latent  heat  of  evaporation.  The  "  heat  in  the 
water  "  referred  to  in  the  steam  tables  is  h  ;  the  "  heat  in  the  steam  "  is 
Hj  also  called  the  total  heat. 

361.  Factor  of  Evaporation.  In  order  to  compare  the  total  expen- 
ditures of  heat  for  producing  saturated  steam  under  unlike  condi- 
tions, we  must  know  the  temperature  T,  other  than  32°  F.  (Art. 
359),  at  which  the  water  is  received,  and  the  pressure  p  at  which 


PRESSURE-TEMPERATURE  205 

steam  is  formed  ;  for  as  T  increases,  h  decreases ;  and  as  p  increases, 
H  increases.  This  is  of  much  importance  in  comparing  the  results 
of  steam  boiler  trials.  At  14.7  Ib.  (atmospheric)  pressure,  for  ex- 
ample, with  water  initially  at  the  boiling  point,  212°  F.,  h  =  0  and 
J3"=£  =  970.4  (from  the  table,  p.  247).  These  are  the  conditions 
adopted  as  standard,  and  with  which  actual  evaporative  performances 
are  compared.  Evaporation  under  these  conditions  is  described  as 
being 

From  (a  feed  water  temperature  of)  and  at  (a  pressure  correspond- 
ing to  the  temperature  of)  212°  F. 

Thus,  for  p  =  200,  we  find  L  =  843.2  and  h  =  354.9 ;  and  if  the  tem- 
perature of  the  water  is  initially  190°  F.,  corresponding  to  the  heat 
contents  of  157.9  B.  t.  u., 

H=  L  +  (354.9  -  157.9)=  843.2  +  197  =  1040.2. 

The  ratio  of  the  total  heat  actually  utilized  for  evaporation  to  that 
necessary  "from  and  at  212°  F."  is  called  the  factor  of  evaporation. 
In  this  instance,  it  has  the  value  1040.2  -r-  970.4  =  1.07.  Generally, 
if  L,  h  refer  to  the  assigned  pressure,  and  hQ  is  the  heat  correspond- 
ing to  the  assigned  temperature  of  the  feed  water,  then  the  factor  of 

evaporation  is 

F=  [Z  +  (7i  -  J0)]  --  970.4. 

362.  Pressure-temperature  Relation.  Regnault  gave,  as  the  result  of  his  ex- 
haustive experiments,  thirteen  temperatures  corresponding  to  known  pressures 
at  saturation.  These  range  from  —  32°  C.  to  220°  C.  He  expressed  the  relation 
by  four  formulas  (Art.  19);  and  no  less  than  fifty  formulas  have  since  been 
devised,  representing  more  or  less  accurately  the  same  experiments.  The  deter- 
minations made  by  Holborn  and  Henning  (16)  agree  closely  with  those  of  Reg- 
nault ;  as  do  those  by  Wiebe  (19)  and  Thiesen  and  Scheel  (20)  at  temperatures 
below  the  atmospheric  boiling  point.  There  is  no  satisfactory  data  at  tempera- 
tures exceeding  500°  F. 

The  steam  table  shows  that,  beginning  at  32°  F.,  the  pressure  rises  with  the 
temperature,  at  first  slowly  and  afterward  much  more  rapidly.  It  is  for  this 
reason  that  two  pressure-temperature  curves,  with  different  pressures  scales,  have 
been  used  in  Fig.  159,  the  low-scale  curve  being  used  for  low  pressures.  If  the 
high-scale  curve  were  extended  downward,  it  would  be  difficult  to  ascertain  accu- 
rately the  pressure  changes  below  atmospheric  for  small  differences  of  tempera- 
ture. The  fact  that  slight  increases  of  temperature  accompany  large  increases  of 
pressure  in  the  working  part  of  the  range  seems  fatal  to  the  development  of  the 
engine  using  saturated  steam,  the  high  temperature  of  heat  absorption  shown  by 


206  APPLIED  THERMODYNAMICS 

Carnot  to  be  essential  to  efficiency  being  unattainable  without  the  use  of  pressures 
mechanically  objectionable. 

A  recent  formula  for  the  relation  between  pressure  and  temperature  is  (Power, 
March  8,  1910)  *  =  200,*  -  101, 

in  which  t  is  the  Fahrenheit  temperature  and  p  the  pressure  in  pounds  per  square 
inch. 

363.  Pressure  and  Volume.     Fairbairn  and  Tate  ascertained  experimentally 
in  1860  the  relation  between  pressure  and  volume  at  a  few  points ;  some  experi- 
ments were  made  by  Him;  and  Battelli  has  reported  results  which  have  been 
examined  by  Tumlirz  (21).     More  recent  experiments  by  Knoblauch,  Linde,  and 
Klebe  (1905)  (22)  give  the  formula 

pv  =  0.5962  T-p(l  +  0.0014 p)  f150'30^000  -  0.0833\ 

in  which  p  is  in  pounds  per  square  inch,  v  in  cubic  feet  per  pound,  and  jTin  degrees 
absolute.  This  may  be  compared  with  Wood's  formula  (23), 

f,  =  0.6732  r-^5. 

A  simple  empirical  formula  is  that  of  Rankine,  PFT6  =  constant,  or  that  of  Zeuner, 
pj/i.0646  _  constant.  These  forms  of  expression  must  not  be  confused  with  the 
PVn  =  c  equation  for  various  poly  tropic  paths.  An  indirect  method  of  determin- 
ing the  volume  of  saturated  steam  is  to  observe  the  value  of  some  thermal  prop- 
erty, like  the  latent  heat,  per  pound  and  per  cubic  foot,  at  the  same  pressure. 

The  incompleteness  of  experimental  determinations,  with  the  diffi- 
culty in  all  cases  of  ensuring  experimental  accuracy,  have  led  to  the  use  of 
analytical  methods  (Art.  368)  for  computing  the  specific  volume.  The 
values  obtained  agree  closely  with  those  of  Knoblauch,  Linde,  and  Klebe. 

364.  Wet  Steam.     Even  when  saturated  steam  is  separated  from 
the  mass  of  water  from  which  it  has  been  produced,  it  nearly  always 
contains  traces  of  water  in  suspension.     The  presence  of  this  water 
produces  what  is  described  as  wet  steam,  the  wetness  being  an  indi- 
cation of  incomplete  evaporation.     Superheated  steam,   of  course, 
cannot  be  wet.     Wet  steam  is  still  saturated  steam  (Art.  356) ;  the 
temperature  and  density  of  the  steam  are  not  affected  by  the  pres- 
ence of  water. 

The  suspended  water  must  be  at  the  same  temperature  as  the 
steam ;  it  therefore  contains,  per  pound,  adopting  the  symbols  of 
Art.  359,  h  units  of  heat.  In  the  total  mixture  of  steam  and  water, 
then,  the  proportion  of  steam  being  x,  we  write  for  L,  xL ;  for  r,  xr ; 
for  e,  xe  ;  for  E,  xr  +  h  ;  while,  h  remaining  unchanged,  H  =  h  +  xL. 


FORMATION   OF   STEAM  207 

The  factor  of  evaporation  (Art.  361),  wetness  considered,  must  be 
correspondingly  reduced  ;  it  is  F  =  [xL  +  Qi—  A0)]  H-  970.4. 

The  specific  volume  of  wet  steam  is  Ww=  V+x  (  W—  V)  =xZ+  V, 
where  Z=  W-  V.  For  dry  steam,  x=  1,  and  Ww  =  V+  (  W-  V)  =  W. 
The  error  involved  in  assuming  Ww=  xWis  usually  inconsiderable, 
since  the  value  of  V  is  comparatively  small. 

365.  Limits  of  Existence  of  Saturated  Steam.     In  Fig.  160,  let 
ordinates  represent  temperatures,  and  abscissas,  volumes.     Then  ab 
is  a  line  representing  possible  condi-      T 

tions  of  water  as  to  these  two  proper- 
ties, which  may  be  readily  plotted  if 
the  specific  volumes  at  various  tem- 
peratures are  known;  and  cd  is  a 
similar  line  for  steam,  plotted  from  the 
values  of  Wand  t  in  the  steam  table. 
The  lines  ab  and  cd  show  a  tendency 
to  meet  (Art.  379).  The  curve  cd  is  FIG.  ieo.  Arts.  365, 366, 379.  -  Paths 

of  Steam  Formation. 

called  the  curve  of  saturation,  or  of  con- 
stant steam  weight ;  it  represents  all  possible  conditions  of  constant 
weight  of  steam,  remaining  saturated.  It  is  not  a  path,  although 
the  line  ab  is  (Art.  363).  States  along  ab  are  those  of  liquid;  the 
area  bade  includes  all  wet  saturated  states ;  along  dc,  the  steam  is 
dry  and  saturated;  to  the  right  of  dc,  areas  include  superheated 
states. 

366.  Path  during  Evaporation.     Starting  at  32°,  the  path  of  the 
substance  during  heating  and  evaporation  at  constant  pressure  would 
be  any  of  a  series  of  lines  aef,  ahi,  etc.     The  curve  ab  is  sometimes 
called  the  locus  of  boiling  points.     If  superheating  at  constant  pres- 
sure occur  after  evaporation,  then  (assuming  Charles'  law  to  hold) 
the  paths  will  continue   as  fg,  ij,  straight  lines  converging  at   0. 
For  a  saturated  vapor,  wet  or  dry,  the  isothermal  can  only  be  a  straight 
line  of  constant  pressure. 

367.  Entropy  Diagram.     Figure  161  reproduces  Fig.  160  on  the 
entropy  plane.     The  line  ab  represents  the  heating  of  the  water  at 
constant  pressure.     Since  the  specific  heat  is  slightly  variable,  the 


208 


APPLIED  THERMODYNAMICS 


increase  of  entropy  must  be  computed  for  small  differences  of  tem- 
perature. The  more  complete  steam  tables  give  the  entropy  at  various 
boiling  points,  measured  above  32°.  Let  evaporation  occur  when  the 


o          o 


M      e  p  t 

FIG.  161.    Arts.  367,  369-373,  376,  379,  386,  426.  —  The  Steam  Dome. 


temperature  is  Tb.  The  increase  of  entropy  from  the  point  b  (since 
the  temperature  is  constant  during  the  formation  of  steam  at  constant 
pressure)  is  simply  L  -f-  (Tb  +  459.6),  which  is  laid  off  as  be.  Other 
points  being  similarly  obtained,  the  saturation  curve  cd  is  drawn. 
The  paths  from  liquid  at  32°  to  dry  saturated  steam  are  abc,  a  VN, 
aUS,  etc. 

The  factor  of  evaporation  may  be  readily  illustrated.  Let  the  area 
e  USf  represent  Lm,  the  heat  necessary  to  evaporate  one  pound  from  and 
at  212°  F.  The  area  gjbch  represents  the  heat  necessary  to  evaporate  one 
pound  at  a  pressure  b  from  a  feed-water  temperature  j.  The  factor  of 
evaporation  is  gjbch  -r-  e  USf.  For  wet  steam  at  the  pressure  b,  it  is,  for 
example,  gjbik  -H  eUSf. 

368.  Specific  Volumes:  Analytical  Method.  This  was  developed  by 
Clapeyron  in  1834.  In  Fig.  162,  let  abed  represent  a  Carnot  cycle  in 
which  steam  is  the  working  substance  and  the  range  of  temperatures  is 
dT.  Let  the  substance  be  liquid  along  da  and  dry  saturated  vapor  along  be. 


VOLUME   OF   VAPOR 


209 


The  heat  area  abfe  is  L;  the  work  area  abed  is  (L  -T-  T)dT.  In  Fig.  163, 
let  abed  represent  the  corresponding  work  area  on  the  pv  diagram.  Since 
the  range  of  temperatures  is  only  dT,  the  range  of  pressures  may  be 


FIGS.  162  and  163.    Arts.  368,  406,  603.  —  Specific  Volumes  by  Clapeyron's  Method. 

taken  as  c?P;  whence  the  area  abed  in  Fig.  163  is  dP(W-  F),  where  W 
is  the  volume  along  6c,  and  V  that  along  ad.  This  area  must  by  the  first 
law  of  thermodynamics  equal  (778  L  -H  T)dT;  whence 


Thus,  if  we  know  the  specific  volume  of  the  liquid,  and  the  latent  heat 
of  vaporization,  at  a  given  temperature,  we  have  only  to  determine  the 

•j  fjj 

differential  coefficient  —  in  order  to  compute  the  specific  volume  of  the 

vapor.  The  value  of  this  coefficient  may  be  approximately  estimated  from 
the  steam  table  ;  or  may  be  accurately  ascertained  when  any  correct  formula 
for  relation  between  P  and  T  is  given.  The  advantage  of  this  indirect 
method  for  ascertaining  specific  volumes  arises  from  the  accuracy  of 
experimental  determinations  of  T,  L,  and  P. 

369.  Entropy  Lines.  In  Fig.  161,  let  ab  be  the  water  line,  cd 
the  saturation  curve;  then  since  the  horizontal  distance  between 
these  lines  at  any  absolute  temperature  T  is  equal  to  L  +  T,  we 
deduce  that,  for  steam  only  partially  dry,  the  gain  of  heat  in  passing 
from  the  water  line  toward  cd  being  xL  instead  of  L,  the  gain  of 
entropy  is  xL  -=-  T  instead  of  L  -r-  T.  If  on  be  and  ad  we  lay  off  bi 
and  al  =  x  •  be  and  x  •  ad,  respectively,  we  have  two  points  on  the 
constant  dryness  curve  il,  along  which  the  proportion  of  dryness  is  x. 
Additional  points  will  fully  determine  the  curve.  The  additional 
curves  zn,  pq,  etc.,  are  similarly  plotted  for  various  values  of  x,  all 
of  the  horizontal  intercepts  between  ab  and  cd  being  divided  in  the 
same  proportions  by  any  one  of  these  curves. 


210  APPLIED  THERMODYNAMICS 

370.  Constant  Heat  Curves.     Let  the  total"  heat  at  o  be  H.     To 
find  the  state  at  the  temperature  be,  at  which  the  total  heat  may  also 
equal  H,  we  remember   that   for   wet   steam   H=  h  -\-  xL,    whence 
x  =  (H—  h)-t-  L  =  lp  -r-  be.     Additional  points  thus  determined  for 
this  and  other  assigned  values  of   H  give   the  constant  total  heat 
curves  ojt?,  wr,  etc.    The  total  heat  of  saturated  vapor  is  not,  however, 
a  cardinal  property  (Art.   10).     The  state  points  on  this  diagram 
determine  the  heat  contents  only  on  the  assumption  that  heat  has 
been  absorbed  at  constant  pressure ;   along  such  paths  as  abc,  aUSt 
aVN,  etc. 

371.  Negative  Specific  Heat.     If  steam  passes  from  o  to  r,  Fig.  161, 
heat  is  absorbed  (area  sort)  while  the  temperature  decreases.     Since  the  satu- 
ration curve  slopes  constantly  downward  toward  the  right,  the  specific  heat 
of  steam  kept  saturated  is  therefore  negative.     The  specific  heat  of  a  vapor 
can  be  positive  only  when  the  saturation  curve  slopes  downward  to  the  left, 
like  cu,  as  in  the  case,  for  example,  of  the  vapor  of  ether  (Fig.  315).     The 
conclusion  that  the  specific   heat   of   saturated   steam   is   negative  was 
reached  independently  by  Rankine  aud  Clausius  in  1850.     It  was  experi- 
mentally  verified   by  Him   in   1862   and   by  Cazin  in  1866  (24).     The 
physical  significance  is  simply  that  when  the  temperature  of  dry  saturated 
steam  is  increased  adiabatically,  it  becomes  superheated ;    heat   must   be 
abstracted  to  keep  it  saturated.     On  the  other  hand,  when  dry  saturated 
steam    expands,    the    temperature    falling,   it    tends    to    condense,    and 
heat  must  be  supplied  to  keep  it  dry.     If  steam  at  c,  Fig.  161,  having 
been  formed  at  constant  pressure,  works  along  the  saturation  curve  to  N, 
its  heat  contents  are  not  the  same  as  if  it  had  been  formed  along  aVN, 
but  are  greater,  being  greater  also  thau  the  "heat  contents"  at  c. 

372.  Liquefaction  during  Expansion.     If  saturated  steam  expand  adia- 
batically from  c,  Fig.  161,  it  will  at  v  have  become  10  per  cent  wet.     If 
its  temperature  increase  adiabatically  from  v,  it  will  at  c  have  become 
dry.   If  the  adiabatic  path  then  continue,  the  steam  will  become  superheated. 
Generally  speaking,  liquefaction  accompanies    expansion  and  drying   or 
superheating  occurs  during  compression.    If  the  steam  is  very  wet  to  begin 
with,  say  at  the  state  x,  compression  may,  however,  cause  liquefaction,  and 
expansion  may  lead  to  drying.     Water  expanding  adiabatically  (path  bz) 
becomes  partially  vaporized.     Vapors  may  be  divided  into  two   classes, 
depending  upon  whether  they  liquefy  or  dry  during  adiabatic  expansion 
under  ordinary  conditions  of  initial  dryness.     At  usual  stages  of  dryness 
and  temperature,  steam  liquefies  during  expansion,  while  ether  becomes 
dryer,  or  superheated. 


INTERNAL  ENERGY   OF   VAPOR 


211 


373.  Inversion.  Figure  161  shows  that  when  x  is  about  0.5  the  constant  dry- 
ness  lines  change  their  direction  of  curvature,  so  that  it  is  possible  for  a  single 
adiabatic  like  DE  to  twice  cut  the  same  dryness  curve ;  x  may  therefore  have  the 
same  value  at  the  beginning  and  end  of  expansion,  as  at  D  and  E.  Further,  it 
may  be  possible  to  draw  an  adiabatic  which  is  tangent  to  the  dryness  curve  at  .4. 
Adiabatic  expansion  below  A  tends  to  liquefy  the  steam  ;  above  .4,  it  tends  to  dry 
it.  During  expansion  along  the  dryness  curve  below  A,  the  specific  heat  is  nega- 
tive; above  .4,  it  is  positive.  By  finding  other  points  like  A,  as  F,  G,  on  similar 
constant  dryness  curves,  a  line  BA  may  be  drawn,  which  is  called  the  zero  line  or 
line  of  inversion.  During  expansion  along  the  dryness  lines,  the  specific  heat 
becomes  zero  at  their  intersection  with  AB,  where  they  become  tangent  to  the 
adiabatics.  If  the  line  AB  be  projected  so  as  to  meet  the  extended  saturation 
curve  rfc,  the  point  of  intersection  is  the  temperature  of  inversion.  There  is  no 
temperature  of  inversion  for  dry  steam  (Art.  379),  the  saturation  curve  reaching 
an  upper  limit  before  attaining  a  vertical  direction. 


374.  Internal  Energy.  In  Fig.  164,  let  2  be  the  state  point  of  a  wet  vapor. 
Lay  off  2  4  vertically,  equal  to  (  T  -H-  L)  (L  -  r).  Then  1  2  4  3  (3  4  being  drawn 
horizontally  and  1  3  vertically)  is  equal  to 

12x24-^  •  I(£-r)=*(L-r). 

This  quantity  is  equal  to  the  external  work  of 
vaporization  =  xe,  which  is  accordingly  repre- 
sented by  the  area  1243.  The  irregular 
area  651347  then  represents  the  addition 
of  internal  energy,  6518  having  been  ex- 
pended in  heating  the  water,  and  834  7=xr 
being  the  disgregation  work  of  vaporization. 


FIG.  164.    Art.  374.  —  Internal  Energy 
and  External  Work. 


375.  External  Work.  Let  MN,  Fig.  165,  be  any  path  in  the  saturated  region. 
The  heat  absorbed  is  mMNn.  Construct  Afcba,  Nfed,  as  in  Art.  374.  The  inter- 
nal energy  has  increased  from  Oabcm  to  Odefn,  the 
amount  of  increase  being  adefnmcb.  This  is  greater 
than  the  amount  of  heat  absorbed,  by  deiMcba  —  iNJ, 
which  difference  consequently  measures  the  external 
work  done  upon  the  substance.  Along  some  such  curve 
as  A!  Y,  it  will  be  found  that  external  work  has  been 
done  by  the  substance. 


o 

FIG.   1(55.      Art.   375. —In- 
ternal Energy  of  Steam. 


376.  The  Entropy  Diagram  as  a  Steam  Table.  In 
Fig.  161,  let  the  state  point  be  H.  We  have  T=  HI, 
from  which  P  may  be  found.  HJ  is  made  equal  to  (7*  -4-  L}(L  —  r),  whence 
Oa  VKJI  =  E  and  VH.TK  =  xe.  Also  x  =  VH  -*•  VN,  the  entropy  measured  from 
the  water  line  is  VH,  the  momentary  specific  heat  of  the  water  along  the  dif- 
ferential path  JL  is  gjLM  +  Tj;  xL  =  PVHI,  xr  =  KJIP,  h  =  OaVP,  and 
H  =  Oa  VHI.  The  specific  volume  is  still  to  be  considered. 


212 


APPLIED  THERMODYNAMICS 


377.  Constant  Volume  Lines.     In  Fig.  166,  let  JA  be  the  water 
line,  BG-  the  saturation  curve,  and  let  vertical  distances  below  ON 
represent  specific  volumes.     Let  xs  equal  the  volume  of  boiling  water, 

sensibly  constant,  and  of  comparatively 
small  numerical  value,  giving  the  line  ss. 
From  any  point  B  on  the  saturation 
curve,  draw  BD  vertically,  making  CD 
represent  by  its  length  the  specific  volume 
at  B.  Draw  BA  horizontally,  and  AE 
vertically,  and  connect  the  points  U&ndD. 
Then  ED  shows  the  relation  of  volume  of 
vapor  and  entropy  of  vapor,  along  AB, 
the  two  increasing  in  arithmetical  ratio. 
Find  the  similar  lines  of  relation  KL  and 
HF  for  the  temperature  lines  JI  and  YGr. 
FIG.  166.  Art.  377.  -  Constant  Draw  the  constant  volume  line  TD,  and 

Volume  Lines. 

find    the    points   on   the    entropy    plane 

w,  v,  B,  corresponding  to  t,  u,  D.     The  line  of  constant  volume  wB 
may  then  be  drawn,  with  similar  lines  for  other  specific  volumes,  qz, 
etc.     The  plotting  of  such  lines  on  the  entropy  plane  permits  of  the 
use   of   this    diagram    for    obtaining 
specific  volumes  (see  Fig.  175). 

378.  Transfer  of  Vapor  States.    In 
Fig.  167,  we  have  a  single  represen- 
tation of  the  four  coordinate  planes 
pt,  tn,  nv,  and  p v.     Let  ss  be  the  line    ^ 
of  water  volumes,  ab  and  efihe  satura- 
tion curve,  Cd  the  pressure-tempera- 
ture curve  (Art.   362),    and    Op   the 
water  line.     To  transfer  points  a,  b  on 
the  saturation  curve  from  the  pv  to  the 

tn  plane,  we  have  only  to  draw  aC, 

Ce,  bd,  and  df.     To  transfer  points 

like  i,  I,  representing  wet  states,  we 

first  find  the  vn  lines  qh  and  rg  as  in  Art.  377,  and  then  project 

ij,jk,  lm,  and  mn  (25). 


FIG.  167.    Art.  378.  —  Transfer  of 
Vapor  States. 


CRITICAL  TEMPERATURE  213 

Consider  any  point  t  on  the  pv  plane.  By  drawing  tu  and  uv  we 
find  the  vertical  location  of  this  point  in  the  tn  plane.  Draw  wA  and 
xB,  making  zB  equal  to  the  specific  volume  of  vapor  at  x  (equal  to 
EF  on  the  p v  plane).  Draw  AB  and  project  t  to  c.  Projecting  this 
last  point  upward,  we  have  D  as  the  required  point  on  the  entropy 
plane. 

379.  Critical  Temperature.  The  water  curve  and  the  curve  of  saturation 
in  Figs.  160  and  161  show  a  tendency  to  meet  at  their  upper  extremities. 
Assuming  that  they  meet,  what  are  the  physical  conditions  at  the  critical 
temperature  existing  at  the  point  of  intersection  ?  It  is  evident  that  here 
L  =  0,  r  =  0,  and  e  =  0.  The  substance  would  pass  immediately  from  the 
liquid  to  the  superheated  condition ;  there  would  be  no  intermediate  state 
of  saturation.  No  external  work  would  be  done  during  evaporation,  and, 
conversely,  no  expenditure  of  external  work  could  cause  liquefaction.  A 
vapor  cannot  be  liquefied,  when  above  its  critical  temperature,  by  any 
pressure  whatsoever.  The  density  of  the  liquid  is  here  the  same  as  that 
of  the  vapor :  the  two  states  cannot  be  distinguished.  The  pressure  re- 
quired to  liquefy  a  vapor  increases  as  the  critical  temperature  is  approached 
(moving  upward)  (Arts.  358, 360)  ;  that  necessary  at  the  critical  temperature 
is  called  the  critical  pressure.  It  is  the  vapor  pressure  corresponding  to  the 
temperature  at  that  point.  The  volume  at  the  intersection  of  the  saturation 
curve  and  the  liquid  line  is  called  the  critical  volume.  The  "  specific  heat 
of  the  liquid  "  at  the  critical  temperature  is  infinity. 

The  critical  temperature  of  carbon  dioxide  is  88.5°  F.  This  substance  is 
sometimes  used  as  the  working  fluid  in  refrigerating  machines,  particularly  on 
shipboard.  It  cannot  be  used  in  the  tropics,  however,  since  the  available  supplies 
of  cooling  water  have  there  a  temperature  of  more  than  88.5°  F.,  making  it  im- 
possible to  liquefy  the  vapor.  The  carbon  dioxide  contained  in  the  microscopic 
cells  of  certain  minerals,  particularly  the  topaz,  has  been  found  to  be  in  the  critical 
condition,  a  line  of  demarcation  being  evident,  when  cooling  was  produced,  and 
disappearing  with  violent  frothing  when  the  temperature  again  rose.  Here  the 
substance  is  under  critical  pressure ;  it  necessarily  condenses  with  lowering  of 
temperature,  but  cannot  remain  condensed  at  temperatures  above  88.5°  F.  Ave- 
narius  has  conducted  experiments  on  a  large  scale  with  ether,  carbon  disulphide, 
chloride  of  carbon,  and  acetone,  noting  a  peculiar  coloration  at  the  critical  point  (26). 

For  steam,  Regnault's  formula  for  H  (Art.  360),  if  we  accept  the  approximation 
h  =  t  —  32°,  would  give  L  =  II  —  h  =  1113.94  —  0.695 /,  which  becomes  zero  when 
t  =  1603°  F.  Davis'  formula  (Art.  360)  (likewise  not  intended  to  apply  to  temper- 
atures above  about  400°  F.)  makes  L  =  0  when  t  =  1709°  F.  The  critical  tempera- 
ture for  steam  has  been  experimentally  ascertained  to  be  actually  much  lower,  the 
best  value  being  about  689°  F.  (27).  Many  of  the  important  vapors  have  been 
studied  in  this  direction  by  Andrews. 


214 


APPLIED  THERMODYNAMICS 


380.  Physical  States.     We  may  now  distinguish  between  the  gaseous 
conditions,  including  the  states  of  saturated  vapor,  superheated  vapor,  and 
true  gas.      A  saturated  vapor,  which  may  be  either  dry  or  wet,  is  a  gaseous 
substance  at  its  maximum  density  for  the  given  temperature  or  pressure ; 
and  belotv  the  critical  temperature.     A  superheated  vapor  is  a  gaseous  sub- 
stance at  other  than  maximum  density  whose  temperature  is  either  less 
than,  or  does  not  greatly  exceed,  the  critical  temperature.     At  higher  tempera^ 
tares,  the  substance  becomes  a  true  gas.     All  imperfect  gases  may  be  regarded 
as  superheated  vapors. 

Air,  one  of  the  most  nearly  perfect  gases,  shows  some  deviations  from  Boyle's  law 
at  pressures  not  exceeding  2500  Ib.  per  square  inch.  Other  substances  show  far  more 
marked  deviations.  In  Fig.  168,  QP  is  an  equilateral  hyperbola.  The  isothermal s 

for  air  at  various  temperatures  centi- 
grade are  shown  above.  The  lower 
curves  are  isothermals  for  carbon  di- 
oxide, as  determined  by  Andrews  (28). 
They  depart  widely  from  the  perfect 
gas  isothermal,  PQ.  The  dotted  lines 
show  the  liquid  curve  and  the  satura- 
tion curve,  running  together  at  a,  at  the 
critical  temperature.  There  is  an  evi- 
dent increase  in  the  irregularity  of  the 
curves  as  they  approach  the  critical  tem- 
perature (from  above)  and  pass  below 
it.  The  curve  for  21.5°  C.  is  particu- 
larly interesting.  From  b  to  c  it  is  a 
liquid  curve,  the  volume  remaining 

practically  constant  at  constant  temperature  in  spite  of  enormous  changes  of  pres- 
sure. From  b  to  d  it  is  a  nearly  straight  horizontal  line,  like  that  of  any  vapor 
between  the  liquid  and  the  dry  saturated  states ;  while  from  d  to  e  it  approaches 
the  perfect  gas  form,  the  equilateral  hyperbola.  All  of  the  isothermals  change 
their  direction  abruptly  whenever  they  ap- 
proach either  of  the  limit  curves  a/or  ag. 

381.  Other  Paths  of  Steam  Formation. 
The  discussion   has  been  limited  to  the 
formation  of  steam  at  constant  pressure, 
the  method  of  practice.     Steam  might  con- 
ceivably  be  formed  along    any    arbitrary 
path,  as  for  instance  in  a  closed  vessel  at 
constant  volume,  the  pressure  steadily  in- 
creasing.     Since  the   change   of   internal 
energy  of    a   substance  depends  upon  its 

initial  and  final  states  only,  and  ^ot  on  the  intervening  path,  a  change  of  path 
affects  the  external  work  only.  For  formation  at  constant  volume,  the  total  heat 
equals  E,  no  external  work  being  done.  If  in  Fig.  169  water  at  c  could  be  com- 


|75- 
I  70 
05- 
60 
55- 
50 


13.1° 


FIG.  168.    Art.  380.  —Critical  Temperature. 


FIG.  169.    Art.  381.  — Evaporation  at 
Constant  Volume. 


SUPERHEATED   STEAM  215 

pletely  evaporated  along  en  at  constant  volume,  the  area  acnd  would  represent  the 
addition  of  internal  energy  and  the  total  heat  received.  If  the  process  be  at  con- 
stant pressure,  along  cbn,  the  area  acbn  d  represents  the  total  heat  received  and  the 
area  cbn  represents  the  external  work  done. 

382.  Vapor  Isodynamic.  A  saturated  vapor  contains  heat  above  32°  F.  equal 
to  h  +  r  +  e  ;  or,  at  some  other  state,  to  hl  +  rx  +  er  If  the  two  states  are  isody- 
nainic  (Art.  83),  h.  +  r  =  h±  +  rv  a  condition  which  is  impossible  if  at  both  states 
the  steam  be  dry.  If  the  steam  be  wet  at  both  states,  h  +  xr  =  ftt  -f  x^rr  Let  p, 
pv  v  be  given  ;  and  let  it  be  required  to  find  vv  the  notation  being  as  in  Art.  364. 

We  have  xl  =  —  :  —  1,  all  of  these  quantities  being  known  or  readily  ascertain- 

able.     Then 


i 
Ifx  =  1.0,  the  steam  being  dry  at  one  state,  xl  =  h  +  r  ~  hi  and 


ri 

Substitution  of  numerical  values  then  shows  that  if  p  exceed  pi,  v  is  less  than  vi  ; 
i.e.  the  curve  slopes  upward  to  the  left  on  the  pv  diagram  :  and  x  is  less  than 
xr  The  curve  is  less  "  steep  "  than  the  saturation  curve.  Steam  cannot  be  worked 
isodynamically  and  remain  dry;  each  isodynamic  curve  meets  the  saturation  curve 
at  a  single  point. 

SUPERHEATED  STEAM 

383.  Properties  :  Specific  Heat.  In  comparatively  recent  years,  superheated 
steam  has  become  of  engineering  importance  in  application  to  reciprocating  en- 
gines and  turbines  and  in  locomotive  practice. 

Since  superheated  steam  exists  at  a  temperature  exceeding  that  of  saturation, 
it  is  important  to  know  the  specific  heat  for  the  range  of  superheating.  The  first 
determination  was  by  Regnault  (1862),  who  obtained  as  mean  values  k  =  0.4805, 
I  =  0.346,  y  =  1.39.  Fenner  found  I  to  be  variable,  ranging  from  0.341  to  0.351. 
Hirn,  at  a  later  date,  concluded  that  its  value  must  vary  with  the  temperature. 
Weyrauch  (29),  who  devoted  himself  to  this  subject  from  1876  to  1904,  finally 
concluded  that  the  value  of  k  increased  both  with  the  pressure  and  with  the 
amount  of  superheating  (range  of  temperature  above  saturation),  basing  this  con- 
clusion on  his  own  observations  as  collated  with  those  of  Regnault,  Hirn,  Zeuner, 
Mallard  and  Le  Chatelier,  Sarrau  and  Veille,  and  Langen.  Rankine  presented  a 
demonstration  (now  admitted  to  be  fallacious)  that  the  total  heat  of  superheated 
steam  was  independent  of  the  pressure.  At  very  high  temperatures,  the  values 
obtained  by  Mallard  and  Le  Chatelier  in  1883  have  been  generally  accepted  by 
metallurgists,  but  they  do  not  apply  at  temperatures  attained  in  power  engineer- 
ing. A  list  by  Dodge  (30)  of  nineteen  experimental  studies  on  the  subject  shows 
a  fairly  close  agreement  with  Regnault's  value  for  k  at  atmospheric  pressure  and 
approximately  212°  F.  Most  experimenters  have  agreed  that  the  value  increases 
with  the  pressure,  but  the  law  of  variation  with  the  temperature  has  been  in 


216 


APPLIED  THERMODYNAMICS 


doubt.  Holborn's  results  (31)  as  expressed  by  Kutzbach  (32)  would,  if  the  em- 
pirical formula  held,  make  k  increase  with  the  temperature  up  to  a  certain  limit, 
and  then  decrease,  apparently  to  zero. 

384.    Knoblauch   and  Jakob   Experiments.     These  determinations  (33) 
have  attracted  much  attention.     They  were  made  by  electrically  super- 


240   280   320   360   400   440   480   520   SCO   600   C40   680   720 

TEMPERATURE-  DEGREES  FAHRENHEIT 

FIG.  170.    Arts.  384,  421.  —  Specific  Heat  of  Superheated  Steam.    Knoblauch  and 

Jakob  Results. 

heating  the  steam  and  measuring  the  input  of  electrical  energy,  which 
was  afterward  computed  in  terms  of  its  heat  equivalent.  These  experi- 
menters found  that  k  increased  with  the  pressure,  and  (in  general) 


SPECIFIC  HEAT 


217 


decreased  with  the  temperature  up  to  a  certain  point,  afterward  increas- 
ing (a  result  the  reverse  in  this  respect  of  that  reported  by  Holborn). 
Figure  170  shows  the  results  graphically.  Greene  (34)  has  used  these  in 
plotting  the  lines  of  entropy  of  superheat,  as  described  in  Art.  398.  The 
Knoblauch  and  Jakob  values  are  more  widely  used  than  any  others  experi- 
mentally obtained. 

385.  Thomas'  Experiments.  In  these,  the  electrical  method  of  heating 
and  a  careful  system  of  radiation  corrections  were  employed  (35).  The 
conclusion  reached  was  that  k  increases  with  increase  of  pressure  and 
decreases  with  increase  of  temperature.  The  variations  are  greatest  near 
the  saturation  curve.  The  values  given  included  pressures  from  7  to  500  Ib. 


EM  I  I  »l  IfUM  II  til  OM  II  It  I  MM  I  »2  13  I 


FIG.  171.     Arts.  385,  388, 


J,  417,  Prob.  42.  —  Specific  Heat  of  Superheated  Steam. 
Thomas'  Experiments. 


per  square  inch  absolute,  and  superheating  ranging  up  to  270°  F.  The 
entropy  lines  and  total  heat  lines  are  charted  in  Thomas'  report.  Within 
rather  narrow  limits,  the  agreement  is  close  between  these  and  the  Knob- 
lauch and  Jakob  experiments.  The  reasons  for  disagreement  outside 
these  limits  have  been  scrutinized  by  Heck  (36),  who  has  presented  a 
table  of  the  properties  of  superheated  steam,  based  on  these  and  other  data. 
The  experiments  may  be  so  readily  duplicated  that  there  is  every  reason 
for  deferring  final  tabulating  until  a  full  set  of  confirmatory  values  shall 
have  been  obtained.  Figure  171  shows  the  Thomas  results  graphically. 

386.    Total  Heat.     As  superheated  steam  is  almost  invariably  formed 
at  constant  pressure,  the  path  of  formation  resembles  abcW,  Fig.  161,  ab 


218  APPLIED  THERMODYNAMICS 

being  the  water  line  and  cd  the  saturation  curve.  Its  total  heat  is  then 
Hc  +  k(T—t),  where  T,  t  refer  to  the  temperatures  at  W  and  c.  If  we 
take  Kegnault's  value  for  He,  1081.94  +  0.305 1  (Art.  360),  then,  using 
k  =  0.4805,  we  find  the  total  heat  of  superheated  steam  to  be  1081.94  — 
0.1755 1  -f-  0.4805  T.  A  purely  empirical  formula,  in  which  P  is  the  pres- 
sure in  pounds  per  square  foot,  is  H=  0.4805(T- 10.37  P0-25)  +  857.2. 
For  accurate  calculations,  the  total  heat  must  be  obtained  by  using  correct 
mean  values  for  k  during  successive  short  intervals  of  temperature  between 
t  and  T. 

387.  Variations  of  k.     Dodge  (37)  has  pointed  out  a  satisfactory  method 
for  computing  the  law  of  variation  of  the  specific  heat.     Steam  is  passed 
through  a  small  orifice  so  as  to  produce  a  constant  reduction  in  a  constant 
pressure.     It  is  superheated  on  both  sides  the  orifice ;  but,  the  heat  con- 
tents remaining  constant  during  the  throttling  operation,  the  temperature 
changes.     Let  the  initial  pressure  be  p,  the  final  pressure  p^     Let  one 
observation  give  for  an  initial  temperature  t,  a  final  temperature  ^;  and 
let  a  second  observation  give  for  an  initial  temperature  T,  a  final  tempera- 
ture TK     Let  the  corresponding  total  heat  contents  be  h,  h^  H,  H^     Then 
h-H=  kp(t  -  T)  and  7^  —  H±  =  k  fa  -  T^.     But  h  =  Uly  H=  H^  whence 

X.  /    rp 

h  —  H—  hl—  HI  and  -*-  =  J l  •   If  we  know  the  mean  value  of  k  for  any 

Kp  t  —  JL 

given  range  of  temperature,  we  may  then  ascertain  the  mean  value  for  a 
series  of  ranges  at  various  pressures. 

388.  Davis'  Computation  of  H-     The  customary  method  of  deter- 
mining k  has  been  by  measuring  the  amount  of  heat  necessarily  added 
to  saturated  steam  in  order  to  produce  an  observed  increase  of  tem- 
perature.    Unfortunately,  the  value  of  If  for  saturated  steam  has 
not  been  known  with  satisfactory  accuracy ;  it  is  therefore  inade- 
quate to  measure  the  total  heat  in  superheated  steam  for  comparison 
with  that  in  saturated  steam  at  the  same  pressure.     Davis  has  shown 
(17)  that  since  slight  errors  in  the  value  of  H  lead  to  large  errors 
in  that  of  &,  the  reverse  computation — using  known  values  of  k  to 
determine  H —  must  be  extremely  accurate ;   so  far  so,  that  while 
additional  determinations  of  the  specific  heat  are  in  themselves  to  be 
desired,  such  determinations  cannot  be  expected  to  seriously  modify 
values  of  H  as  now  computed. 

The  basis  of  the  computation  is,  as  in  Art.  387,  the  expansion  of 
superheated  steam  through  a  non-conducting  nozzle,  with  reduction 


VALUE  OF  H  219 

of  temperature.  Assume,  for  example,  that  steam  at  38  Ib.  pres- 
sure and  300°  F.  expands  to  atmospheric  pressure,  the  temperature 
becoming  286°  F.  The  total  heat  before  throttling  we  may  call 
He  =  Hb-\-kl(fTc—Tb)^  in  which  Hb  is  the  total  heat  of  saturated 
steam  at  38  Ib.  pressure,  Tc  =  300°  F.,  and  Tb  is  the  temperature  of 
saturated  steam  at  38  Ib.  pressure,  or  264.2°  F.  After  throttling, 
similarly,  Hd  =  He  +  k^Td  —  Te),  in  which  He  is  the  total  heat  of 
saturated  steam  at  atmospheric  pressure,  Te  is  its  temperature 
(212°  F.),  and  Td  is  286°  F.  Now  Hd=  Hc,  and  He  =  1150.4  ;  while 
from  Fig.  171  we  find  kl  =  0.57  and  k2  =  0.52  ;  whence 

Hb  =  -  0.57(300  -  264.2)  +  1150.4  +  0.52(286  -  212)=  1168.47. 

The  formula  given  by  Davis  as  a  result  of  the  study  of  various 
throttling  experiments  may  be  found  in  Art.  360.  The  total  heat 
of  saturated  steam  at  some  one  pressure  (e.g.  atmospheric)  must  be 
known. 

•  A  simple  formula  (that  of  Smith),  which  expresses  the  Davis  results  with  an 
accuracy  of  1  per  cent,  between  70°  and  500°,  was  given  in  Power,  February  8,  1910. 

it  is  fl 


1620  -  t 
t  being  the  Fahrenheit  temperature. 

389.  Factor  of  Evaporation.     The  computation  of  factors  of  evapora- 
tion must  often  include  the  effect  of  superheat.     The  total  heat  of  super- 
heated steam  —  which  we  may  call  H.  —  may  be  obtained  by  one  of  the 
methods  described  in  Art.  386.     If  7<0  is  the  heat  in  the  water  as  sup- 
plied, the  heat  expended  is  H,  —  h$  and  the  factor  of  evaporation  is 

(H.-hd-s-  970.4. 

390.  Characteristic  Equation.     Zeuner   derives  as  a  working  formula, 
agreeing  with  Hirn's  experiments  on  specific  volume  (38), 

PV=  0.64901  T-  22.5819  P0-25, 

in  which  P  is  in  pounds  per  square  inch,  V  in  cubic  feet  per  pound,  and 
T  in  degrees  absolute  Fahrenheit.  This  applies  closely  to  saturated  as 
well  as  to  superheated  steam,  if  dry.  Using  the  same  notation,  Tumlirz 
gives  (39)  from  Battelli's  experiments, 

PV=  0.594  T-  0.00178  P. 

The  formulas  of  Knoblauch,  Linde  and  Jakob,  and  of  Wood,  both  given 
in  Art.  363,  may  also  be  applied  to  superheated  steam,  if  not  too  highly 


220  APPLIED  THERMODYNAMICS 

superheated.  At  very  high  temperatures,  steam  behaves  like  a  perfect  gas, 
following  closely  the  law  PV=  RT.  Since  the  values  of  R  for  gases  are 
inversely  proportional  to  their  densities,  we  find  R  for  steam  to  be  85.8. 

391.  Adiabatic  Equation.     Using  the  value  just  obtained  for  R,  and  Regnault's 
constant  value  0.4805  for  fc,  we  find  y  =  1.298.     The  equation  of  the  adiabatic 
would  then  be  pvl-m  =  c.     This,  like  the  characteristic  equation,  does  not  hold 
for  wide  state   ranges;   a  more   satisfactory  equation  remains  to  be  developed 
(Art.  397).     The  exponential  form  of  expression  gives  merely  an  approximation 
to  the  actual  curve. 

PATHS  OF  VAPORS 

392.  Vapor  Adiabatics.     It  is  obvious  from  Art.  372  that  during 
adiabatic  expansion  of  a  saturated  vapor,  the  condition  of  dryness 

must  change.  We  now  compute  the  equa- 
tion of  the  adiabatic  for  any  vapor.  In 
Fig.  172,  consider  expansion  from  b  to  c. 
Draw  the  isothermals  T,  t.  We  have 


N  m-*-|£S  +  2«nd*-*-»,JLbe- 


/ 
FIG.  172.    Art.  392.— Equa-  ing  the  variable  temperature  along  da.    But 

tion  of  Vapor  Adiabatic.        M  _  m     _A  ^ 


7  TT          7" 

constant   and  equal  to  c,  ^  =  cloge—  +  —  J,  the  desired  equation. 

t        t   J. 

If  the  vapor  be  only  X  dry  at  b,  then 


393.  Applications.    This  equation  may  of  course  be  used  to  derive  the  results 
shown  graphically  in  Art.  373.      For  example,  for  steam  initially  dry,  we  may 
make  X  =  1,  and  it  will  be  always  found  that  xc  is  less  than  1.     To  show  that 
water  expanding  adiabatically  partially  vaporizes,  we  make  X  =  0.     To  determine 
the  condition  under  which  the  dryness  may  be  the  same  after  expansion  as  before 
it,  we  make  x  =  X. 

394.  Approximate  Formulas.     Rankine  found  that  the  adiabatic  might  be 
represented  approximately  by  the  expression, 

PV™  =  constant; 

which  holds  fairly  well  for  limited  ranges  of  pressure  when  the  initial  dryness  is 
1.0,  but  which  gives  a  curve  lying  decidedly  outside  the  true  adiabatic  for  any  con- 
siderable pressure  change.  The  error  is  reduced  as  the  dryness  decreases,  down  to 
a  certain  limit.  Zeuner  found  that  an  exponential  equation  might  be  written  in 


STEAM  ADIABATICS 


221 


the  form  PV*  =  constant,  if  the  value  of  n  were  made  to  depend  upon  the  initial 
dryness.  He  represented  this  by 

n  =  1.035  +  0.100  X, 

for  values  of  X  ranging  from  0.70  to  1.00,  and  found  it  to  lead  to  sufficiently  accu- 
rate results  for  all  usual  expansions.  For  a  compression  from  an  initial  dryness  ar, 
n  =  1.034  +  0.11  x.  Where  the  steam  is  initially  dry,  n  =  1.135  for  expansion  and 
1.144  for  compression.  There  is  seldom  any  good  reason  for  the  use  of  exponential 
formulas  for  steam  adiabatics.  The  relation  between  the  true  adiabatic  and  that 
described  by  the  exponential  equation  is  shown  by  the  curves  of  Fig.  173  after 


0  5  10  ID 

FIG.  173.    Arts.  394,  395.  —  Adiabatic  and  Saturation  Curves. 

Heck  (40).  In  each  of  these  five  sets  of  curves,  the  solid  line  represents  the 
adiabatic,  while  the  short-dotted  lines  are  plotted  from  Zeuner's  equation,  and  the 
long-dotted  lines  represent  the  constant  dryness  curves.  In  I  and  II,  the  two 
adiabatics  apparently  exactly  coincide,  the  values  of  x  being  1.00  and  0.75.  In 
III,  IV,  and  V,  there  is  an  increasing  divergence,  for  x  =  0.50,  0.25  and  0.  Case 
V  is  for  the  liquid,  to  which  no  such  formula  as  those  discussed  could  be  expected 
to  apply. 

395.  Adiabatics  and  Constant  Dryness  Curves.  The  constant  dryness  curves 
I  and  II  in  Fig.  173  fall  above  the  adiabatic,  indicating  that  heat  is  absorbed  during 
expansion  along  the  constant  dryness  line.  Since  the  temperature  falls  during 
expansion,  the  specific  heat  along  these  constant  dryness  curves,  within  the  limits 
shown,  must  necessarily  be  negative,  a  result  otherwise  derived  in  Art.  373.  The 
points  of  tangency  of  these  curves  with  the  corresponding  adiabatics  give  the 
points  of  inversion,  at  which  the  specific  heat  changes  sign. 


222 


APPLIED  THERMODYNAMICS 


396.  External  Work.    The  work  during  adiabatic  expansion  from 
tQ  pv,  assuming  pvn  =  PVn,  is  represented  by  the  formula 

PV-pv 
n-  1 

More  accurately,  remembering  that  the  work  done  equals  the  loss  of 
internal  energy,  we  find  its  value  to  be  H—  h  +  XR  —  xr,  in  which 
H  and  h  denote  the  initial  and  final  heats  of  the  liquid. 

397.  Superheated  Adiabatic.     Three  cases  are  suggested  in  Fig.  174,  paths /m, 
Jk,  de,  the  initially  superheated  vapor  being  either  dry,  wet,  or  superheated  at  the 


FIG.  174.    Art.  397.  —  Steam  Adiabatics. 

end  of  expansion.      If  k  be  the  mean  value  of  the  specific  heat  of  superheated 
steam  for  the  range  of  temperatures  in  each  case,  then 


for/*,  c  loge       +        +  k  loge    JL  = 


for  de,  c  log,     6  +    f  +  k,  loge       =    f+  *»  log*      . 


398.  Entropy  Lines  for  Superheat.  Many  problems  in  superheated 
steam  are  conveniently  solved  by  the  use  of  a  carefully  plotted  entropy 
diagram,  as  shown  in  Fig.  175.*  The  plotting  of  the  curves  within  the 
saturated  limits  has  already  been  explained.  At  the  upper  right-hand 
corner  of  the  diagram  there  appear  constant  pressure  lines  and  constant 
total  heat  curves.  The  former  may  be  plotted  when  we  know  the  mean 
specific  heat  k  at  a  stated  pressure  between  the  temperatures  T  and  t  :  the 


entropy  gained  being  A;  loge— . 

t 


The  lines  of  total  heat  are  determined 


*  This  diagram  is  based  on  saturated  steam  tables  embodying  Regnault's  results,  and 
on  Thomas'  values  for  k ;  it  does  not  agree  with  the  tables  given  on  pages  247,  248.  The 
same  remark  applies  to  Figs.  159  and  177. 


STEAM   ENTROPY  DIAGRAM 


223 


ggggg    * 


01   02   03   0^   Ol5   0^   07   OS   09   \0   11   1  '2   13   M   15   l|6   17   1.8   19 

FIG.  175.     Arts.  377,  398,  401,  411,  417,  516,  Problems. 


224 


APPLIED  THERMODYNAMICS 


by  the  following  method :  —  For  saturated  steam  at  103.38  Ib.  pressure, 
#=1182.6,  77=330°  F.  As  an  approximation,  the  total  heat  of  1200 
B.  t.  u.  will  require  (1200  -  1182.6) -s-  0.4805  =  36.1°  F.  of  superheating. 
For  this  amount  of  superheating  at  100  Ib.  pressure,  the  mean  specific 
heat  is,  according  to  Thomas  (Fig.  171),  0.604 ;  whence  the  rise  in  tem- 
perature is  17.4  -H  0.604  =  28.7°  F.  For  this  range  (second  approxima- 
tion), the  mean  specific  heat  is  0.612,  whence  the  actual  rise  of  temperature 
is  17.4  -f-  0.612  =  28.4°  F.  No  further  approximation  is  necessary ;  the 
amount  of  superheating  at  1200  B.  t.  u.  total  heat  may  be  taken  as  28°  F., 

which  is  laid  off 
vertically  from  the 
point  where  the  satu- 
ration curve  crosses 
the  line  of  330°  F., 
giving  one  point  on 
the  1200  B.  t.  u.  total 
heat  curve. 

A  few  examples 
in  the  application  of 
the  chart  suggest 
themselves.  Assume 
steam  to  be  formed 
at  103.38  Ib.  pres- 
sure ;  required  the 
necessary  amount  of 
superheat  to  be  im- 
parted such  that  the 
steam  shall  be  just 
dry  after  adiabatic 
expansion  to  atmos- 
pheric pressure.  Let 
rs,  Fig.  176,  be  the 

line  of  atmospheric  pressure.  Draw  st  vertically,  intersecting  di',  then 
t  is  the  required  initial  condition.  Along  the  adiabatic  ts,  the  heat  contents 
decrease  from  1300  B.  t.  u.  to  1150.4  B.  t.  u.,  a  loss  of  149.6  B.  t.  u. 

To  find  the  condition  of  a  mixture  of  unequal  weights  of  water  and  super- 
heated steam  after  the  establishment  of  thermal  equilibrium,  the  whole 
operation  being  conducted  at  constant  pressure :  let  the  water,  amounting 
to  10  Ib.,  be  at  r,  Fig.  176.  Its  heat  contents  are  1800  B.  t.  u.  Let  one 
pound  of  steam  be  at  t,  having  the  heat  contents  1300  B.  t.  u.  The  heat 
gained  by  the  water  must  equal  that  lost  by  the  steam  ;  the  final  heat  con- 
tents will  then  be  3100  B.  t.  u.,  or  282  B.  t.  u.  per  pound,  and  the  state 


FIG.  176.    Arts.  398,  399,  401.  —  Entropy  Diagram,  Superheated 
Steam. 


226 


APPLIED  THERMODYNAMICS 


will  be  u,  where  the  temperature  is  312°  F. ;  the  steam  will  have  been 
completely  liquefied. 

We  may  find,  from  the  chart,  the  total  heat  in  steam  (wet,  dry,  or 
superheated)  at  any  temperature,  the  quality  and  heat  contents  after 
adiabatic  expansion  from  any  initial  to  any  final  state,  and  the  specific 
volume  of  saturated  steam  at  any  temperature  and  dryness. 

399.  The  Mollier  Heat  Chart.  This  is  a  variant  on  the  temperature 
entropy  diagram,  in  a  form  rather  more  convenient  for  some  purposes.  It 
has  been  developed  by  Thomas  (41)  to  cover  his  experiments  in  the 
superheated  region,  as  in  Fig.  177.  In  this  diagram,  the  vertical  coordi- 
nate is  entropy ;  and  the  horizontal,  total  heat.  The  constant  heat  lines 
are  thus  vertical,  while  adiabatics  are  horizontal.  The  saturation  curve 
is  inclined  upward  to  the  right,  and  is  concave  toward  the  left.  Lines  of 
constant  pressure  are  nearly  continuous  through  the  saturated  and  super- 
heated regions.  The  quality  lines  follow  the  curvature  of  the  saturation 
line.  The  temperature  lines  in  the  superheated  region  are  almost  vertical. 
It  should  be  remembered  that  the  "  total  heat "  thus  used  as  a  coordinate 
is  nevertheless  not  a  cardinal  property.  The  "total  heat"  at  t,  Fig.  176, 
for  example,  is  that  quantity  of  heat  which  would  have  been  imparted 
had  water  at  32°  F.  been  converted  into  superheated  steam  at  constant 
pressure. 

The  total  heat-pressure  diagram  (Fig.  185)  is  a  diagram  in  which  the  coordi- 


1550 


PRESSURE.    POUNDS  PER    SQUARE  INCH 

555    3  S3  3     I  8 


750 


80    100  120    140  160  180   200  220   210   260  280  300    320   340  360  380  400    420  440  460  480 
SATURATED  STEAM   TEMPERATURE.  DEGREES  F. 

FIG.  185.    Art.  399,  Problems.  —Total  Heat-pressure  Diagram. 


MATHEMATICAL  THERMODYNAMICS  227 

nates  are  total  heat  above  32°  F.  and  pressure;  it  usually  includes  curves  of 
(a)  constant  volume,  (&)  constant  dryness,  (c)  constant  temperature.  Vertical  lines 
show  the  loss  or  gain  of  heat  corresponding  to  stated  changes  of  volume  or  quality 
at  constant  pressure.  Horizontal  lines  show  the  change  in  pressure,  volume,  and 
quality  of  steam  resulting  from  throttling  (Art.  387).  This  diagram  is  a  use- 
ful supplement  to  that  of  Mollier. 

VAPORS  IN  GENERAL 

400.    Analytical  Method:   Mathematical  Thermodynamics.    An  expression 
for  the  volume  of  any  saturated  vapor  was  derived  in  Art.  368: 


Where  the  specific  volume  is  known  by  experiment,  this  equation  may  be  used  for 
computing  the  latent  heat.  A  general  method  of  deriving  this  and  certain  related 
expressions  is  now  to  be  described.  Let  a  mixture  of  x  Ib.  of  dry  vapor  with 
(1  —  x)  Ib.  of  liquid  receive  heat,  dQ.  Then  * 

dQ  =  kxdT  +  c  (1  -  x)  dT  +  Ldx, 

in  which  k  is  the  "specific  heat"  of  the  continually  dry  vapor,  L  the  latent  heat 
of  evaporation,  and  c  the  specific  heat  of  the  liquid.  If  P.V  are  the  pressure  and 
volume,  and  E  the  internal  energy,  in  foot-pounds,  of  the  mixture,  then 

dQ  =  PdV^  dE  =  kxdT  +  c(l-x)dT  +  Ldx,  whence 

<78 

dE  =  778  [fcc  +  c  (1  -  *)]  dT  +  778  Ldx  -  PdV. 
Now  V=  (/)  T,  x',  whence  dV  =  |T  dT  +  |?  dx,  whence 

61  OX 

dE  =  778  [kx  +  c  (1  -  *)]  dT  +  77SLdx  -  P  ^  dT  -  P  ^-dx 

ol  ox 


Moreover,  E  =  (/)  !T,  ar,  whence 


(all  properties  excepting  Tr  and  x  being  functions  of  T  only). 

The  volume,  V,  may  be  written  xu  +  i?,  where  v  is  the  volume  of  the  liquid  and 

8K 

u  the  increase  of  volume  during  vaporization.     This  gives  oV  =  vox  or  —  =  u. 


228  APPLIED  THERMODYNAMICS 

On  T7  SkSTT" 

Also,  since  V=  (/)  T,  x,  f——  =  ^=,t  and  equation  (A)  becomes 

0  1OX        OXO  1 


dL  +  r  -  k  -    U    dP>  (B^ 

dT+  "778dT 

Now   if  the   heat    is   absorbed    along    any  reversible    path,     -Q  =  dN,    or 
IN      kxdT+c(l  -x}dT  +  Ldx  =  kx  +  c(l  -  aQ  dT  ,    L  , 

r 

But.  *- 


rpdL  j- 

k-c_      dT~ 
T  T2 

^f  +  c-*  =  f'  (C) 

which  may  be  combined  with  (B),  giving 

™TdP=  "y'aS 

401.  Computation  of  Properties.  Equation  (D),  as  thus  derived,  or  as  obtained 
in  Art.  369,  may  be  used  to  compute  either  the  latent  heat  or  the  volume  of  any 
vapor  when  the  other  of  these  properties  and  the  relation  of  temperature  and  pres- 
sure is  known.  The  specific  heat  of  the  saturated  vapor  may  be  obtained  from 
(C)  ;  the  temperature  of  inversion  is  reached  when  the  specific  heat  changes  sign. 
For  steam,  if  L  =  1113.94  -  0.695 1  (Art.  379),  where  t  is  in  degrees  F.,  or 

1113.94  -  0.695(7T  -  459.6)  where  T  is  the  absolute  temperature:   —  T=  -  0.695. 
Also  c  =  1 ;  whence,  from  equation  (C),  k  =  0.305  —  — ,  which  equals  zero  when 


- 
T=  1433°  absolute.*    At  212°  F.,  k  =  0.305  -  =  ^  1.135.     This  may  be  roughly 

checked  from  Fig.  175.  In  Fig.  176,  consider  the  path  sb  from  212°  F.  to  157°  F., 
and  from  n  =  1.735  to  n  =  1.835  (Fig.  175).  The  average  height  of  the  area  csbe 

O19    _j_    1  K"7 

representing  the  heat  absorbed  is  459.6  +  -   '— =  644.1 ;  whence,  the  area  is 

644.1(1.835  -  1.735)  =  64.41  B.  t.  u.,  and  the  mean  specific  heat  between  *  and  b  is 
64.41  H-  (212  —  157)  =  1.176.  The  properties  of  the  volatile  vapors  used  in  refriger- 
ation are  to  some  extent  known  only  by  computations  of  this  sort.  When  once 
the  pressure-temperature  relation  and  the  characteristic  equation  are  ascertained  by 
experiment,  the  other  properties  follow. 

*  This  would  be  the  temperature  of  inversion  of  dry  steam  if  the  formula  for  L  held  : 
but  L  becomes  zero  at  689°  F.  (Art.  379),  and  the  saturation  curve  for  steam  slopes  downward 
toward  the  right  throughout  its  entire  extent.  For  the  dry  vapors  of  chloroform  and  ben- 
zine, there  exist  known  temperatures  of  inversion. 


VAPORS  IN   GENERAL  229 

402.  Engineering  Vapors.     The  properties  of  the  vapors  of  steam,   carbon 
dioxide,  ammonia,  sulphur  dioxide,  ether,  alcohol,  acetone,  carbon  disulphide,  carbon 
tetrachloride,  and  chloroform  have  all  been  more  or  less  thoroughly  studied.     The 
first  five  are  of  considerable  importance.     For  ether,  alcohol,  chloroform,  carbon  disul- 
phide, carbon  tetrachloride,  and  acetone,  Zeuner  has  tabulated  the  pressure,  tempera- 
ture, volume,  total  heat,  latent  heat,  heat  of  the  liquid,  and  internal  and  external 
work  of  vaporization,  in  both  French  and   English  units  (42),  on  the  basis  of 
Regnault's  experiments.     The  properties  of  these  substances  as  given  in  Peabody's 
"Steam  Tables"  (1890)  are  reproduced  from  Zeuner,  excepting  that  the  values 
-  273.7  and  426.7  are  used  instead  of  -  273.0  and  424.0  for  the  location  of  the 
absolute    zero    centigrade    and    the   centigrade   mechanical  equivalent  of    heat, 
respectively.     Peabody's  tables  for  these  vapors  are  in  French  units  only.     Wood 
has  derived  expressions  for  the  properties  of  these  six  vapors,  but  has  not  tabulated 
their  values  (43).     Rankine  (44)  has  tabulated  the  pressure,  latent  heat,  and  density 
of  ether,  per  cubic  foot,  in  English  units,  from  Regnault's  data.     For  carbon  dioxide, 
the  experimental  results  of  Andrews,  Cailletet  and  Hautefeuille,  Cailletet  and 
Mathias    (45),  and,  finally,  Amagat  (46),  have  been  collated  by  Mollier,  whose 
table  (47)  of  the  properties  of  this  vapor  has  been  reproduced  and  extended,  in 
French  and  English  units,  by  Zeuner  (48).     The  vapor  tables  appended  to  Chapter 
XVIII,  it  will  be  noted,  are  based  on  those  of  Zeuner.   The  entropy  diagrams  for  am- 
monia, sulphur  dioxide,  and  carbon  dioxide,  Figs.  314-316,  have  the  same  foundation. 

403.  Ammonia.     Anhydrous   ammonia,  largely  used   in   refrigerating 
machines,  was  first  studied  by  Regnault,  who  obtained  the  relation 

log  p  =  8.4079-?^, 

in  which  p  is  in  pounds  per  square  foot  and  t  is  the  absolute  temperature. 
A  "characteristic  equation"  between  p,  v,  and  t  was  derived  by  Ledoux 
(49)  and  employed  by  Zeuner  to  permit  of  the  computation  of  V,  L,  e,  r 
and  the  specific  heat  of  the  liquid  (the  last  having  recently  been  deter- 
mined experimentally  (50)).  The  results  thus  derived  were  tabulated  by 
Zeuner  (51)  for  temperatures  below  32°  F.  ;  while  for  higher  temperatures 
he  uses  the  experimental  values  of  Dietrici  (52).  Peabody's  table  (53), 
also  derived  from  Ledoux,  uses  his  values  for  temperatures  exceeding 
32°  F. ;  Zeuner  regards  Ledoux's  values  in  this  region  as  unreliable. 
Peabody's  table  is  in  French  units ;  Zeuner's  is  in  both  French  and  Eng- 
lish units.  The  latent  heat  of  evaporation  has  been  experimentally  de- 
termined by  Regnault  (54)  and  Yon  Strombeck  (55).  The  specific  volume 
of  the  vapor  at  —  26.4°  F.  and  atmospheric  pressure  is  17.51  cu.  ft. ;  that  of 
the  liquid  is  0.025 ;  whence  from  equation  (D),  Art.  400, 


230  APPLIED  THERMODYNAMICS 

the  value  of  —  being  obtained  by  differentiating  Eegnault's  equation, 

above  given.     From  a  study  of  Regnault's  experiments,  Wood  has  derived 
the  characteristic  equation, 

PF  =  91_  16920 
T  ~ 


which  is  the  basis  of  his  table  of  the  properties  of  ammonia  vapor  (56). 
Wood's  table  agrees  quite  closely  with  Zeuner's,  as  to  the  relation  between 
pressure  and  temperature  ;  but  his  value  of  L  is  much  less  variable.  For 
temperatures  below  0°  C.,  the  specific  volumes  given  by  Wood  are  rather 
less  than  those  by  Zeuner;  for  higher  temperatures,  the  volumes  vary 
less.  Zeuner's  table  must  be  regarded  as  probably  more  reliable.  The 
specific  heat  (0.508)  and  the  density  (0.597,  when  air  =  1.0)  of  the  super- 
heated vapor  have  been  determined  by  experiment. 

404.  Sulphur  Dioxide.  The  specific  heat  of  the  superheated  vapor  is  given  by 
Regnault  as  0.15438  (57).  The  specific  volume,  as  compared  with  that  of  air,  is 
2.23  (58).  The  specific  volume  of  the  liquid  is  0.0007  (59)  ;  its  specific  heat  is 
approximately  0.4.  A  characteristic  equation  for  the  saturated  vapor  has  been 
derived  from  Regnault's  experiments  : 

PV  =  26.4  T-  184  P°-22; 

in  which  P  is  in  pounds  per  square  foot,  Fin  cubic  feet  per  pound,  and  T  in  abso- 
lute degrees.  The  relation  between  pressure  and  temperature  has  been  studied  by 
Regnault,  Sajotschewski,  Blumcke,  and  Miller.  Regnault's  observations  were 
made  between  -  40°  and  149°  F.  ;  Miller's,  between  68  and  211°  F.  ;  a  table  repre- 
senting the  combined  results  has  been  given  by  Miller  (60).  In  the  usual  form 
of  the  general  equation, 

log  p  =  a  —  b  dn  —  cen, 

the  values  given  by  Peabody  for  pressures  in  pounds  per  square  inch  are  (61) 
a  =  3.9527847,  log  b  =  0.4792425,  log  d  =  1.9984994,  log  c  =  1.1659562,  loge  = 
1.99293890,  n  =  18.4  +  Fahrenheit  temperature.  The  specific  volumes,  determined 
by  the  characteristic  equation  and  the  pressure-temperature  formula,  permit  of  the 
computation  of  the  latent  heat  from  equation  (D),  Art.  400.  An  empirical  formula 
for  this  property  is  L  =  176  -  0.27(<  -  32),  in  which  t  is  the  Fahrenheit  tempera- 
ture. The  experimental  results  of  Cailletet  and  Mathias,  and  of  Mathias  alone  (62), 
have  led  to  the  tables  of  Zeuner  (63).  Peabody,  following  Ledoux's  analysis,  has 
also  tabulated  the  properties  in  French  units.  Wood  (64)  has  independently  com- 
puted the  properties  in  both  French  and  English  units.  Comparing  Wood's,  Zeu- 
ner's, and  Peabody's  tables,  Zeuner's  values  for  L  and  V  are  both  less  than  those  of 
Peabody.  At  0°  F.,  he  makes  L  less  than  does  Wood,  departing  even  more  widely 
than  the  latter  from  Jacobus'  experimental  results  (65)  ;  at  30°  F.,  his  value  of  L  is 
greater  than  Wood's,  and  at  104°  F.,  it  is  again  less.  The  tabulated  values  of  the 
specific  volumes  differ  correspondingly.  Zeuner's  table  may  be  regarded  as  sus- 


STEAM   PLANT   CYCLE 


231 


tained  by  the  experiments  of  Cailletet  and  Mathias,  but  the  lack  of  concordance 
with  the  experimental  results  of  Jacobus  remains  to  be  explained. 

405.  Steam  at  Low  Temperatures.     Ordinary  tables  do  not  give  the  proper- 
ties of  water  vapor  for  temperatures  lower  than  those  corresponding  to  the  abso- 
lute pressures  reached  in  steam  engineering.     Zeuner  has,   however,   tabulated 
them  for  temperatures  down  to  —  4°  F.  (66). 

STEAM  CYCLES 

406.  The  Carnot  Cycle  for  Steam.     This  is  shown  in  Figs.  163, 
179.     The  efficiency  of  the  cycle  abed  may  be  read  from  the  entropy 

T-t 


diagram  as 


T 


The  external 


work  done  per  pound  of  steam 

T  —  t 
is   L — - — ;   or  if  the  steam  at  b 

T-t 


is  wet,  it  is  xL 


T 


If   the 


N 

FIG.  179.    Art.  406.  —Carnot  Cycle  for  Steam. 


fluid  at  the  beginning  of  the 
cycle  (point  a)  is  wet  steam 
instead  of  water,  the  dryness 
being  #0,  then  the  work  per 
pound  of  steam  is  L(x  —  #0) 

m j. 

In  the  cycle  first  discussed,  in  order  that  the  final  adiabatic 

compression  may  bring  the  substance  back  to  its  initially  dry  state  at 
a,  such  compression  must  begin  at  d,  where  the  dryness  is  md  -*-  mn. 

The  Carnot  cycle  is  impracticable 
with  steam ;  the  substance  at  d  is 
mostly  liquid,  and  cannot  be  raised 
in  temperature  by  compression. 
What  is  actually  done  is  to  allow 
condensation  along  ed  to  be  com- 
pleted, and  then  to  warm  the  liquid 
or  its  equivalent  along  ma  by  trans- 
mission of  heat  from  an  external 
source.  This,  of  course,  lowers 
the  efficiency. 

407.    The  Steam  Power  Plant.     The  cycle  is  then  not  completed  in 
the  cylinder  of  the  engine.     In  Fig.  180,  let  the  substance  at  d  be 


CON3ENSER 


FIG.  180.     Arts.  407,  408,  410,  412,  413.  — 
The  Steam  Power  Plant. 


232 


APPLIED  THERMODYNAMICS 


cold  water,  either  that  resulting  from  the  action  of  the  condenser 
on  the  fluid  which  has  passed  through  the  engine,  or  an  external 
supply.  This  water  is  now  delivered  by  the  feed  pump  to  the  boiler, 
in  which  its  temperature  and  pressure  become  those  along  ab.  The 
work  done  by  the  feed  pump  per  pound  of  fluid  is  that  of  raising 
unit  weight  of  the  liquid  against  a  head  equivalent  to  the  pressure ; 
or,  what  is  the  same  thing,  the  product  of  the  specific  volume  of  the 
water  by  the  range  in  pressure,  in  pounds  per  square  foot.  From 
a  to  b  the  substance  is  in  the  boiler,  being  changed  from  water  to 
steam.  Along  be,  it  is  expanding  in  the  cylinder;  along  cd  it  is 
being  liquefied  in  the  condenser  or  being  discharged  to  the  atmos- 
phere. In  the  former  case,  the  resulting  liquid  reaches  the  feed 
pump  at  d.  In  the  latter,  a  fresh  supply  of  liquid  is  taken  in  at  d, 
but  this  may  be  thermally  equivalent  to  the  liquid  resulting  from 
atmospheric  exhaust  along  cd.  (See  footnote,  Art.  502.)  The  four 

organs,  feed  pump,  boiler,  cylinder, 
and  condenser,  are  those  essential  in 
a  steam  power  plant.  The  cycle  rep- 
resents the  changes  undergone  by 
the  fluid  in  its  passage  through  them. 

408.  Clausius  Cycle.  The  cycle 
of  Fig.  180,  worked  without  adiabatic 
compression,  is  known  as  that  of 
Clausius.  Its  entropy  diagram  is 
shown  as  debc  in  Fig.  181,  that  of 
the  corresponding  Carnot  cycle  being 
dhbc.  The  Carnot  efficiency  is  obviously  greater  than  that  of  the 
Clausius  cycle.  For  wet  steam  the  corresponding  cycles"  are  dekl 
and  dhkl. 

409.    Efficiency.     In  Fig.  181,  cycle  debc,  the  efficiency  is 
debc        idej  +  jebK—  idcK  _  he  —  hd  +  Lb  —  xcLf 


FIG.  181. 


Arts.  408-413.  —  Steam 
Cycles. 


idebK  idej  -\-jebK 


he-h 


But  xc=  dc-r-  df= 


T> 


T, 


if  the  specific  heat  of  the 


RANKINE   CYCLE  233 

liquid  be  unity.     Then  letting  T,  L  refer  to  the  state  6,  and  £,  I  to 
the  state  c,  the  efficiency  is 


T-t  +  L  T-t  +  L 

which  is  determined  solely  by  the  temperature  limits  T  and  t.     For 
steam  initially  wet,  the  efficiency  is 


T-  t  +  XL 

410.  Work  Area.     In  Figs.  180,  181,  we  have 

W=  Wab  +  Wbc  -  Wcd  -  Wda 

=  [Pb(?b  -  v*)]  +  (A  +  rb  ~  hd  -  x<rf)  -  Or^cO/  ~  vdy]  -  0, 

ignoring  the  small  amount  of  work  done  by  the  feed  pump  in  forcing 
the  liquid  into  the  boiler.  But  pb(vb  —  v0)  =  eb  and  p<xc(vf  —  vd)=xcef 

(Art.  359),  whence 

W=  he  +  Lb-hd-  xjjj, 

a  result  identical  with  the  numerator  of  the  first  expression  in  Art. 
409. 

411.  Rankine  Cycle.     The  cycle  debgq,  Fig.  181,  abgqd,  Fig.  180, 
is  known  as  that  of  Rankine  (67).     It  differs  from  that  of  Clausius 
merely  in  that  expansion  is  incomplete,  the  "toe"  gcq,  Fig.  180, 
being  cut  off  by  the  limiting  cylinder  volume  line  gq.     This  is  the 
ideal  cycle  nearest  which  actual  steam  engines  work.     The  line  gq  in 
Fig.  181  is  plotted  as  a  line  of  constant  volume  (Art.  377).     The 
efficiency  is  obviously  less  than  that  of  the  Clausius  cycle  ;  it  is 

ebggd  _  Wab  +  W^  -  Wqd  (Fig.  180) 
idebK  he-hd  +  Lb 

=  [Pb(vb  -  Pa)]  +  (A,  +  rb  -  hr  -  xffrs)  -  \_pqXq(vf  -  ^)]  . 
he  —  hd  +  Lb 

The  values  of  h^  xg^  r^  xq,  depend  upon  the  limiting  volume  vg  —  vv, 
and  may  be  most  readily  ascertained  by  inspecting  Fig.  175.  The 
computation  of  these  properties  resolves  itself  into  the  problem  :  given 


234  APPLIED  THERMODYNAMICS 

the  initial  state,  to  find  the  temperature  after  adiabatic  expansion  to  a 
given  volume.     We  have 


T      L 


ns  —  nr      ns  —  nr         L<  -r-  Tr 
whence 

77         r 
^ 


in  which  vg,  Te,  Lb  are  given,  vr  =  0.017,  and  0,,  ia  are  functions  of 
2^,  the  value  of  which  is  to  be  ascertained. 

412.  Non-expansive  Cycle.     This  appears  as  debt,  Fig.  181  ;  and  abed,  Fig.  180. 
No  expansion  occurs  ;   work  is  done  only  as  steam  is  evaporated  or  condensed. 
The  efficiency  is  (Fig.  181) 

debt   =  Wab  -  Wed  (Fig.  180)  =  Pb(vb  -  gg)  -  Pt  fo  -  Vd) 
idebK  he-hd+Lb  he  -  hd  +  Lb 

This  is  the  least  efficient  of  the  cycles  considered. 

413.  Pambour  Cycle.     The  cycle  debf,  Fig.  181,  represents  the  operation  of  a 
plant  in  which  the  steam  remains  dry  throughout  expansion.     It  is  called  the 
Pambour  cycle.     Expansion  may  be  incomplete,  giving  such  a  diagram  as  debug. 
Let  abed  in  Fig.  180  represent  debf  in  Fig.  181.     The  efficiency  is 

external  work  done  external  work  done 


gross  heat  absorbed      heat  rejected  +  external  work  done 
=       Wab  +  Wbn  -  Wcd      =       Pb(vb  -  ru)  +  16(  /)»?»- 


at,  +  Wbc  -  Wcd     Lf  +  p 

in  which  the  saturation  curve  bf  may  be  represented  by  the  formula  pv^t  =  con- 
stant (Art.  363).     A  second  method  for  computing  the  efficiency  is  as  follows: 

the  area  debf—  \    —dT,in  which  T  and  t  are  the  temperatures  along  eb  and  df 
respectively,  and  L=(f)T=  1433  -  0.695  T  (Art.  379).     This  gives 

debf=  1433  log.-  -  0.695(7*  -  0, 
and  the  efficiency  is 

delf  _        de¥  14831og.f-0.895(r-0 


!'">      1433  log,^-  0.695(7--  Q+L, 


SUPERHEATED   CYCLES 


235 


!T 


The  two  computations  will  not  precisely  agree,  because  the  exponent  £|  does  not 
exactly  represent  the  saturation  curve,  nor  does  the  formula  for  L  in  terms  of  T 
hold  rigorously. 

Of  the  whole  amount  of  heat  supplied,  the  portion  Kbfv  was  added 
during  expansion,  as  by  a  steam  jacket  (Art.  439).  To  ascertain  this 
amount,  we  have 

heat  added  by  jacket 

=  whole  heat  supplied  —  heat  present  at  beginning  of  expansion 

=  1433  log,  -  -  0.695  (I7-  t)  +  Lf-  he  +  hd  -  Lb. 

The  efficiency  is  apparently  less  than  that  of  the  Clausius  cycle  (Fig. 
181).  In  practice,  however,  steam  jacketing  increases  the  efficiency  of 
engines,  for  reasons  which  will  appear  (Art.  439). 

x 

414.  Cycles  with  Superheat.  As  in  Art.  397,  three  cases  are  pos- 
sible. Figure  182  shows  the  Clausius  cycles  debvw,  debyf,  debzAf, 
in  which  the  steam  is  respectively  wet,  dry,  and  superheated  at  the 
end  of  expansion.  To  appreciate 
the  gain  in  efficiency  due  to  super- 
heat, compare  the  first 'of  these 
cycles,  not  with  the  dry  steam 
Clausius  cycle  dehe,  but  with  the 
superior  Carnot  cycle  dhbc.  If  the 
path  of  superheating  were  bC,  the 
efficiency  would  be  unchanged ; 
the  actual  path  is  bx,  and  the  work 
area  bx  C  is  gained  at  100  per  cent 
efficiency.  The  cycle  debxw  is 
thus  more  efficient  than  the  Car- 
not cycle  dhbc,  and  consequently 
still  more  efficient  than  the  Clausius  cycle  debc.  It  is  not  more 
efficient  than  a  Camot  cycle  through  its  own  temperature  limits. 

The  cycle  debyf  shows  a  further  gain  in  efficiency,  the  work  area 
added  at  100  per  cent  effectiveness  being  byE.  The  cycle  debzAf 
shows  a  still  greater  addition  of  this  desirable  work  area,  but  a  loss  of 
area  AfB  now  appears.  Maximum  efficiency  appears  to  be  secured 
with  such  a  cycle  as  the  second  of  those  considered,  in  which  the 
steam  is  about  dry  at  the  end  of  expansion.  The  Carnot  formula 


FIG    182. 


Art.  414.  —Cycles  with 
Superheat. 


236  APPLIED  THERMODYNAMICS 

suggests  the  desirability  of  a  high  upper  temperature,  and  superheating 
leads  to  this  ;  but  when  superheating  is  carried  so  far  as  to  appreciably 
raise  the  temperature  of  heat  emission,  as  in  the  cycle  debzAf,  the 
efficiency  begins  to  fall. 

415.    Efficiencies.     The  work  areas  of  the  three  cycles  discussed 
may  be  thus  expressed,: 

Wdebxw  =  Hdebxw  =  Hde  +  Heb  -f  ffbx  —  Hwd 

=  he  -hd+Lb  +.*!(  Tx  -  Tb)  -  xwLf  ; 
=  Hdebyf  =  Hde  +  Heb  +  Hby  —  Hfd 


WdebzAf  =  HdebzAf  =  Hdf.  +  Heb  +  Hb2  —  HAf  —  Hfd 

=  he  -hd  +  Lb  +  ks(T2  -  Tb)  -  k,(TA  -  Tf)  -  Lf, 

in  which  kv  &2,  &3,  &4,  refer  to  the  mean  specific  heats  over  the  re- 
spective .pressure  and  temperature  ranges.  The  efficiencies  are 
obtained  by  dividing  these  expressions  by  the  gross  amounts  of  heat 
absorbed.  The  equations  given  in  Art.  397  permit  of  computation 
of  such  quantities  as  are  not  assumed. 

416.  Itemized  External  Work.  The  pressure  and  temperature  at  the 
beginning  of  expansion  being  given,  the  volume  may  be  computed  and 
the  external  work  during  the  reception  of  heat  expressed  in  terms  of 
P  and  F".  The  temperature  or  pressure  at  the  end  of  expansion  being 
given,  the  volume  may  be  computed  and  the  negative  external  work 
during  the  rejection  of  heat  calculated  in  similar  terms.  The  whole 
work  of  the  cycle,  less  the  algebraic  sum  of  these  two  work  quantities 
(the  feed  pump  work  being  ignored),  equals  the  work  under  the 
adiabatic,  which  may  be  approximately  checked  from  the  formula 

—  ^,  a  suitable  value  being  used  for  n  (Art.  394).     A  second 

approximation  may  be  made  by  taking  the  adiabatic  work  as  equivalent 
to  the  decrease  in  internal  energy,  which  at  any  superheated  state  has 

k 
the  value  h  -f-  r  +  -(T—  f),  T  being  the  actual  temperature,  and  A,  r, 

y 

t  referring  to  the  condition  of  saturated  steam  at  the  stated  pressure. 
The  most  simple  method  of  obtaining  the  total  work  of  the  cycle  is  to 


COMPARISONS 


237 


read  from  Fig.  177  the  "  total  heat"  values  at  the  beginning  and  end  of 
expansion. 

417.   Comparison  of  Cycles.     In  Fig.  183,  we  have  the  following 
cycles  : 

T 


tpswq 


FIG.  183.    Arts.  417,  441,  442.  —  Seventeen  Steam  Cycles. 


Clausius, 


Rankine, 


with  dry  steam,  dele  (the  corresponding  Carnot 

cycle  being  dhbc*); 
with  wet  steam,  dekl; 
with  dry  steam,  debgq; 
with  wet  steam,  dekJq; 
Non-expansive,  with  dry  steam,  debt ; 

with  wet  steam,  dekK\ 
Pambour,  complete  expansion,  debf; 

incomplete  expansion,  debuq; 
Superheated  to  #,  complete  expansion,  debxw ; 

incomplete  expansion,  debxLuq; 

no  expansion,  debxNp; 
Superheated  to  y,  complete  expansion,  debyf; 

incomplete  expansion,  debyMuq; 

no  expansion,  debyRs ; 

Superheated  to  2,  complete  expansion,  debzAf; 
incomplete  expansion,  debzTuq; 
no  expansion,  debzVw. 


238 


APPLIED  THERMODYNAMICS 


The  lines  f5,  pNx,  sRy,  wVz,  quT,  are  lines  of  constant  volume. 
Superheating  without  expansion  would  be  unwise  on  either  technical 
or  practical  grounds ;  superheating  with  incomplete  expansion  is  the 
condition  of  universal  practice  in  reciprocating  engines.  The 
seventeen  cycles  are  drawn  to  PV  coordinates  in  Fig.  184. 


y   z 


c     w        f 


FIG.  184.     Arts.  417,  423,  424,  517.  —  Seventeen  Steam  Cycles. 

ILLUSTRATIVE  PROBLEM 

To  compare  the  efficiencies,  and  the  cyclic  areas  as  related  to  the  maximum  volume  at- 
tained: let  the  maximum  pressure  be  140  lb.,  the  minimum  pressure  2  lb.,  and  consider 
the  Clausius  cycle  (a)  with  steam  initially  dry,  (/;)  with  steam  initially  90  per  cent 
dry;  the  Rankine  with  initially  dry  steam  and  a  maximum  volume  of  13  cu.  ft., 
the  same  Rankine  with  steam  initially  90  per  cent  dry  ;  the  non-expansive 
with  steam  dry  and  90  per  cent  dry  ;  the  Pambour  (a)  with  complete  expansion 
and  (&)  with  a  maximum  volume  of  13  cu.  ft.  ;  and  the  nine  types  of  superheated 
cycle,  the  steam  being  ;  (a)  96  per  cent  dry,  (b)  dry,  (c)  40°  F.  superheated,  at  the 
end  of  complete  expansion  ;  and  expansion  being  (a)  complete,  (6)  limited  to  a 
maximum  volume  of  13  cu.  ft.,  (c)  eliminated. 

I.    Clausius  cycle.     The  gross  heat  absorbed  is  AHO-£2  +  Z;i40=  324.6  -94.0  +  867.6 

=  1098.2. 
The  dryness  at  the  end  of  expansion  is  dc  -*-  rf/*,  Fig.  183,  =  (ne  —  nd  +  neb)  -*-  ndf 

=  (0.5072  -  0.1749  +  1.0675)  *  1.7431  =  0.803. 
The  heat  rejected  along  cd  is  xcLf  =  0.803  x  1021  =  819.4. 


The  work  done  is  1098.2  -  819.4  =  278.8 B.  t.  u.     The  efficiency  is 

The  efficiency  of  the  corresponding  Carnot  cycle  is 
T^-T,,      353.1-126.15 


1098.2 


=  0.254. 


7'140          353.1+  459.6 


=  0.28. 


II.    Clausius  cycle  with  wet  steam.     The  gross  heat  absorbed  is  A140  —  h2  +  xkLm 

=  324.6  -  94.0+  (0.90  x  867.6)  =  1015.44. 

The  dryness   at    the  end   of   expansion    is   dl  ~  'If—  (ne  —  nd  +  n^)  -f-  wrf/- 
=  (0.5072  -  0.1749  +  0.90  x  1.0675)  -  1.7431  =  '0.741. 


COMPARISONS  239 

The  heat  rejected  along  Id  is  xtLf=  0.741  x  1021  =  756. 
The  work  done  is  1015.44  -  756  =  259.44  B.  t.  u. 

f)-Q   A  A 

Inefficiency  is  ^-^0.^4. 

(It  is  in  all  cases  somewhat  less  than  that  of  the  initially  dry  steam  cycle.) 

III.   Rankine  cycle,  dry  steam.     The  gross  heat  absorbed,  as  in  I,  is  1098.2. 

The  work  along  de,  Fig.  184, is  144  x  138  x  0.017  =  338.5  foot-pounds  (Art.  407); 
along eb  is  144  x  140  x  (Vb  -  0.017)  =  64,300 foot-pounds ; 

(F6  =  3.219) 
along  bg  is  he  +  rb  -  hz  -  xgrg  =  109.76  B.  t.  u. 

(From  Fig.  175,  ^  =  247°  F.,  whence  ^  =  947.4,  Va=  14.52,  x9=  14521001'7 

=  0.895.) 


»Za  La   •*•    Ta 

=  [0.5072 -2.3 (log  Ta  -  Iog491.6)  +  1.0675]  Tg 

1433  -  0.695  Tg 

For  Tg  =  247°  F.  =  706.6°  absolute,  this  equation  gives  xg  =  0.905 ;  a  suffi- 
cient check,  considering  that  Fig.  175  is  based  on  a  different  set  of  values 
than  those  used  in  the  steam  table.     Then  hz  =  215.4,  rg  =  871.6. 
The  work  along  qd  is  Pd(  Vq  -  7d)  =  144  x  2  x  (13  -  0.017)  =  3740  foot- 
pounds. 

The  Me  xork  of  the  cycle  is  64300-338.5-3740  +  109.76  = 

7/8 

The  efficiency  is         "9  =  0.1704- 


IV.   Rankine  cycle,  wet  steam.     The  gross  heat  absorbed  is  as  in  II,  101544. 

The  negative  work  along  de  and  qd  is,  as  in  III,  338.5  +  3740  =  4078.5  foot- 

pounds. 

The  work  along  ek  is  144  x  140  x  0.90(7*  -  0.017)=  57,870  foot-pounds. 
The  work  along  kJ  is  he  +  xkrb  -  hx-  XjrT  =  99.8  B.  t.  u. 
(From  Fig.  175,  tx=  242°  F.,  whence  ^  =  210.3,  rr  =  875.3,  VT=  15.78, 
13-0.017 


15.78  -  0.017 
The  whole  work  of  the  cycle  is  ?7870  ~Q4078'5  +  99.8  =  169.1  B.  t.  u. 


-  The  efficiency  is  '      -  0.1667. 

.i'i 


V.    Non-expansive  cycle,  dry  steam.     The  gross  heat  absorbed,  as  in  I,  is  1098.2. 
The  work  along  de,  as  in  III,  is  338.5  foot-pounds  ; 
along  eb,  as  in  III,  is  64,300  foot-pounds  ; 

along  td  ispd(Vb  -  Fd)  =  144  x  2  x  (3.219  -  0.017)=  922  foot-pounds. 
The  whole  work  of  the  cycle  is 

64,300  -  338.5  -  922  =  63,039.5  foot-pounds  =  81.05  B.  t.  u. 

The  efficiency  is  =  0.074- 


240  APPLIED  THERMODYNAMICS 

VI.  Non-expansive  cycle,  wet  steam.  The  gross  heal  absorbed,  as  in  II,  is  1015.44. 
The  work  along  de,  ek,  as  in  IV,  is  -  338.5  +  57,870  =  57,531.5  foot-pounds. 
The  work  along  Kd  is 

Pd(VK-  0.017)=  144  x  2  x  0.90  x  (3.219  -  0.17)=  829.8  foot-pounds. 
The  whole  work  of  the  cycle  is 

57,531.5  -  829.8  =  56,701.7  foot-pounds  =  73  B.  t.  u. 

>JO   A 

The  efficiency  is  —  -    '—  =  0.0782. 
101o.44 

VII.    Pambour  cycle,  complete  expansion.     The  heat  rejected  is  .£/=  1021.0. 

The  work  along  de,  eb,  as  in  III,  is  —  338.5  +  64300  =  63,961.5  foot-pounds. 
The  work  along  bfis 

**-*W=  U4((HOx3.219)-(2xm.5)\  = 

n  —  1  \  <J  J  —  1  / 

The  work  along  fd  is  Pd(  F,  -  Fd)  =  2  x  144  (173.5  -  0.017)  =  .#0,000  foot- 

pounds. 

The  toAofe  worfc  of  the  cycle  is  63,961.5  +  236,800  -  49,900  =  250,861.5  foot- 
pounds. 

(Otherwise  1433  loge—  -  0.695  (T7  -  0  =  312  B.  t.  u.  =  242,000  foot-pounds 

(Art.  413).) 
Using  a  mean  of  the  two  values  for  the  whole  work,  the  gross  heat  absorbed 


is  +  1021  =  1340  B.  t.  u.  and  the  efficiency  is      24643°      =  0.238. 

The  heat  supplied  by  the  jacket  is  1340  -  1098.2  =  246.8  B.  t.  u. 

VIII.  Pambour  cycle,  incomplete  expansion  (debuq).  In  this  case,  we  cannot 
directly  find  the  heat  rejected,  nor  can  we  obtain  the  work  area  by  inte- 
gration.* From  Fig.  175  (or  from  the  steam  table),  we  find  Tu  =  253.8°  F., 
PM=  31.84.  The  heat  area  under  bu  is  then,  very  nearly, 

T"  +  Tb  (n,  -  n5)=  712'6  +  812-7  (1.6953  -  1.5747)  =  92  B.  t.  u. 

The  whole  heat  absorbed  is  then  1098.2  +  92  =  1190.2  B.  t.  u. 
The  work  along  de,  eb,  as  in  VII,  is  63,961.5  foot-pounds. 
The  work  along  bu  is  144  x  16[(140  x  3.219)-  (31.84  x  13)]  =  85,800  foot- 

pounds. 

The  work  along  qd,  as  in  III,  is  3740  foot-pounds. 
The  whole  work  of  the  cycle  is 

63,961.5  +  85,800  -  3740  =  146,021.5  foot-pounds  =  188.2  B.  t.  u. 

1  QQ  f) 

The  efficiency  is    *  °'w  =  0.1585. 
1190.2 

*  A  satisfactory  solution  may  be  had  by  obtaining  the  area  of  the  cycle  in  two  parts,  a 
horizontal  line  being  drawn  through  u  to  de.  The  upper  part  may  then  be  treated  as  a  com- 
plete-expansion Pambour  cycle  and  the  lower  as  a  non-expansive  cycle.  The  gross  heat 
absorbed  is  equal  to  the  work  of  the  upper  cycle  plus  the  latent  heat  of  vaporization  at  the 
division  temperature  plus  the  difference  of  the  heats  of  liquid  at  the  division  temperature 
and  the  lowest  temperature. 

A  somewhat  similar  treatment  leads  to  a  general  solution  for  any  Rankine  cycle  :  in 
which,  if  the  temperature  at  the  end  of  expansion  be  given,  the  use  of  charts  becomes 
unnecessary. 


COMPARISONS  241 

IX.  Superheated  cycle,  steam  0.96  dry  at  the  end  of  expansion;  complete  expansion; 
cycle  debxw.  We  have  nw  =  nd+  x^n^  =  0.1749  +  (0.96  x  1.7431)  =  1.8449. 
The  state  x(nx  —  nw)  may  now  be  found  either  from  Fig.  175  or  from  the 
superheated  steam  table.  Using  the  last,  we  find  Tx  =  931.  1°F.,/7Z  =  1481.8, 
Vx  =  5.96.  The  whole  heat  absorbed,  measured  above  Td,  is  then 

1481.8  -  94.0  =  1387.8. 

The  heat  rejected  is  xwLf  =  0.96  x  1021  =  981. 
The  external  work  done  is  1387.8  -  981  =  406.8,  and  the  efficiency  is 

406.8 


138T8 

(The  efficiency  of  the  Carnot  cycle  within  the  same  temperature  limits  is 
931.1-  126.15 
931.1  +  459.6 

X.    The  same  superheated  cycle,  with  incomplete  expansion. 
The  whole  heat  absorbed,  as  before,  is  1387.8. 
The  work  done  along  de,  eb,  as  in  III,  is  63,961.5  foot-pounds. 
The  work  done  along  bx  is 

Pi(Vx  ~  J'"0=  144  x  140(5.96  -  3.219)=  55,000  foot-pounds. 
The  work  done  along  xL  is 

=  144  (  (UQx5.se)-  (oi.i  xi3n  =  81tSOO  f<mt_pounds. 

V  0.298  / 


—  1 


(VL  =  13,  PxVxu»  =  PLVL***t  PL  =  i  =  51.1  ;    a    procedure 

which  is,  however,  only  approximately  correct  (Art.  391).) 
The  work  along  qd,  as  in  III,  is  37  40  foot-pounds. 
The  whole  work  of  the  cycle  is 

1.5  +55,000  +  81,500  -  3740  =  196,721.5  foot-pounds  =  253.5  B.  t.  u. 


9x0  x 

The  efficiency  is  ^      ^  =  0.183. 
lob/  .o 

XI.    The  same  superheated  cycle,  worked  non-expansively.     The  gross  heat  absorbed 

is  1387.8. 

The  work  along  de,  eb,  bx,  as  in  X,  is  118,961.5  foot-pounds. 
The  work  along  pd  is  2  x  144  x  (5.96  -  0.017)  =  1716  foot-pounds. 
The  whole  work  of  the  cycle  is 

118,961.5  -  1716  =  117,  245.  5  foot-pounds  =  150.6  B.  t.  u. 


The  efficiency  is  --  =  0.1086. 

.  8 


XII.    Superheated  cycle,  steam  dry  at  the  end  of  expansion,  complete  expansion  ;  cycle 

debyf. 

We  have  ny  =  nf  —  1.918.  This  makes  the  temperature  at  y  above  the 
range  of  our  table.  Figure  171  shows,  however,  that  at  high  tempera- 
tures the  variations  in  the  mean  value  of  k  are  less  marked.  We  may 
perhaps  then  extrapolate  values  in  the  superheated  stearn  table,  giving 
Ty  =  1120.1°  F.,  Hy  =  1573.5,  Vy  =  6.81.  The  whole  heat  absorbed,  above 
Td,  is  then  1573.5  —  94.0  =  1479.5.  The  heat  rejected  is  Lf  =  1021. 


242  APPLIED  THERMODYNAMICS 

The  external  work  done  is  1479.5  —  1021  =  458.5  B.  t.  u.,  and  the  efficiency 


XIII.    Superheated  cycle  as  above,  but  with  incomplete  expansion.     The  gross  heat 

absorbed  is  1479.5. 

The  work  done  along  de,  eb,  as  in  III,  is  63,961.5  foot-pounds. 
The  work  done  along  by  is  144  x  140  x  (6.81  -  3.219)  =  72,200  foot-pounds. 

/6  81  \  lt298 
The  pressure  at  M  is  140  (  —  ^—  )         =  60.3  pounds,  approximately. 


13  / 

The  work  done  along  yM  is  144  f-(140  X  6-81V-(60-3  x  v^\  =  W00  foot. 

pounds,  also  approximately. 

The  work  done  along  qd,  as  in  III,  is  37  4-0  foot-pounds. 
The  whole  work  of  the  cycle  is 

63,961.5  +  72,200  +  81,100  -  3740  =  213,521.5  foot-pounds  =  275  B.  t.  u. 

97^ 

The  efficiency  is     "  ''    „  =  0.187. 


XIV.    Superheated  cycle  as  above,  but  without  expansion.     The  gross  heat  absorbed 

is  1479.5. 

The  z0or&  a/on*/  de,  eb,  by,  as  in  XIII,  is  136,161.5  foot-pounds. 
The  work  along  sd  is  2  x  144  x  (6.81  -  0.017)  =  1952  foot-pounds. 
The  total  work  is  136,161.5  -  1952  -  134,209.5  foot-pounds  =  172.7  B.  t.  u. 

179  7 
' 


The  efficiency  is 


„ 


XV.  Superheated  cycle,  steam  superheated  40°  F.  at  the  end  of  expansion  ;  expan- 
sion complete  ;  cycle  debzAf.  We  have  nA  =  nz  =  1.9486.  A  rather 
doubtful  extrapolation  now  makes  T,  =  1202.1°  F.,  //,  =  1613.4,  V2 
=  7.18.  The  whole  heat  absorbed  is  1613.4  -  94.0  "=  15194-  The  heat  re- 
jected is  HA  =  1133.2.  The  total  tvork  is  1519.4  -  1133.2  =  386.2  B.  t.  «., 

386  "^ 
and  the  efficiency  is  — 


XVI.    The  same  superheated  cycle,  with  incomplete  expansion.     The  pressure  at  T  is 

140  (  —  ;  —  )        =65.3  pounds.     The  work    along   zT  (approximately)  is 
\  13  / 

144  /(140x7.18)-(8S.8xl8)\  =  7^900  foot.pounds.     The  toMe  work  is 

\  0.298  / 

63,961.5  +  [144  x  140  x  (7.18  -  3.219)]  +  73,900  -  3740  =  213,921.5  foot- 

pounds =  275.3  B.  t.  u.,  and  the  efficiency  is     .         =  0.182. 

Iol9.4 

XVII.  The  same  superheated  cycle  without  expansion.  The  total  work  is  63,961.5  -f 
[144  x  140  x  (7.18  -  3.219)]  -  [2  x  144  x  (7.18  -  0.017)]  =  141,701.5  foot- 
pounds =  182.2  B.  t.  M.,  and  the  efficiency  is  0.120S. 

418.    Discussion  of  Results.     The  saturated  steam  cycles  rank  in 
order  of  efficiency  as  follows:  Carnot,  0.28;  Clausius,  with  dry  steam, 


COMPARISONS 


243 


0.254 ;  with  wet  steam,  0.254  (a  greater  percentage  of  initial  wetness 
would  have  perceptibly  reduced  the  efficiency) ;  Pambour,  with  com- 
plete expansion,  0.238  ;  with  incomplete  expansion,  0.1585  ;  Rankine, 
with  dry  steam,  0.1704;  with  wet  steam,  0.1667;  non-expansive,  with 
dry  steam  0.074;  with  wet  steam,  0.0722.  The  economical  impor- 
tance of  using  initially  dry  steam  and  as  much  expansion  as  possible 
is  evident.  The  Pambour  type  of  cycle  has  nothing  to  commend  it, 
the  average  temperature  at  which  heat  is  received  being  lowered. 
The  Rankine  cycle  is  necessarily  one  of  low  efficiency  at  low  expan- 
sion, the  non-expansive  cycle  showing  the  maximum  waste. 

Comparing    the    superheated    cycles,   we   have    the    following 
efficiencies : 


CYCLE 

COMPLETE  EXPANSION 

INCOMPLETE  EXPANSION 

No  EXPANSION 

debxw 

0.293 

0.183 

0.1086 

debyf 
debzAf 

0.31 
0.255 

0.187 
0.182 

0.117 

0.1203 

The  approximations  used  in  solution*  will  not  invalidate  the 
conclusions  (a)  that  superheating  gives  highest  efficiency  when  it  is 
carried  to  such  an  extent  that  the  steam  is  about  dry  at  the  end  of 
complete  expansion;  (5)  that  incomplete  expansion  seriously  re- 
duces the  efficiency  ;  (c)  that  in  a  non-expansive  cycle  the  effi- 
ciency increases  indefinitely  with  the  amount  of  superheating.  As 
a  general  conclusion,  the  economical  development  of  the  steam  en- 
gine seems  to  be  most  easily  possible  by  the  use  of  a  superheated 
cycle  of  the  finally-dry-steam  type,  with  as  much  expansion  as  pos- 
sible. We  shall  discuss  in  Chapter  XIII  what  practical  modifica- 
tions, if  any,  must  be  applied  to  this  conclusion. 

The  limiting  volumes  of  the  various  cycles  are 


Vc  for  the  Carnot,  I,  =  139.3. 

Vl  for  11  =  128.2. 

Vu  =  Vg  for  III,  IV,  VIII,  X,  XIII,  XVI  =  13, 

Vb  for  V  =  3.219. 

Vk  for  VI  =  2.9. 

Vf  for  VII,  XII  =  173.5. 

*  See  footnote,  Problem  53,  page  255. 


Vw  for  IX       =166.5. 
Vx  for  XI       =5.96. 
Vy  for  XIV    =  6.81. 
F4forXV      =186.1. 
Vz  for  XVII  =  7.18. 


241 


APPLIED  THERMODYNAMICS 


The  capacity  of  an  engine  of  given  dimensions  is  proportional  to 


cyclic  area 


-,  which  quotient  has  the  following  values* :  — 


maximum  volume 

Carnot,  temperature  range  x  entropy  range 
=  226.95(1.5747  -  0.1749)=  317.5:  quotient 


317.5 


I.  278.8^139.3  =  2.00.  X. 

II.  259.44 -H  128.2  =  2.015.  XI. 

III.  187. 29 -r- 13  =  14.4.  XII. 

IV.  169.1^13=13.0.  XIII. 
V.  81.05-5-3.219  =  25.1.  XIV. 

VI.  73.0-2.9=25.1.  XV. 

VII.  318^-173.5  =  1.84.  XVI. 

VIII.  188.2-13  =  14.5.  XVII. 

IX.  406.8-^166.5  =  2.445. 


139.3 
253.5 -H  13  =  19.45. 
150.6^-5.96  =  25.3. 
458.5-^-173.5  =  2.65. 
275 -j- 13  =  21.1. 
172.7 -r- 6.81  =25.4. 
386.2-186.1  =  2.075. 
275.3 -5- 13  =  21.1. 
182.2-5-7.18  =  25.5. 


2.29. 


Here  we  find  a  variation  much  greater  than  is  the  case  with  the 
efficiencies ;  but  the  values  may  be  considered  in  three  groups,  the 
first  including  the  five  non-expansive  cycles,  giving  maximum 
capacity  (and  minimum  efficiency)  ;  the  second  including  the  six 
cycles  with  incomplete  expansion,  in  which  the  capacity  varies  from 
13  to  21.1  and  the  efficiency  from  0.1585  to  0.187;  and  the  third 
including  six  cycles  of  maximum  efficiency  but  of  minimum  capacity, 
ranging  from  1.84  to  2.65.  In  this  group,  fortunately,  the  cycle  of 
maximum  efficiency  (XII)  is  also  that  of  maximum  capacity. 

*  The  assumption  of  a  constant  limiting  volume  line  Twg,  Fig.  183,  is  scarcely 
fair  to  the  superheated  steam  cycles.  In  practice,  either  the  ratio  of  expansion  or  the 
amount  of  constant  volume  pressure-drop  at  the  end  of  expansion  is  assumed.  As  the 
first  increases  and  the  second  decreases,  the  economy  increases  and  the  capacity  figure 
decreases.  The  following  table  suggests  that  with  either  an  equal  pressure  drop  or  an 
equal  expansion  ratio  the  efficiencies  of  the  superheated  cycles  would  compare  still 
more  favorably  with  that  of  the  Kankine  :  — 

CYCLES  WITH  INCOMPLETE  EXPANSION 


CYCLE 

•   ... 

RATIO 

Vf 

EXPANSION 

PRESSURE  DROP 

Rankine 

*,* 

vb  = 

13 

-3.219  = 

4.04 

PI- 

Pg =  26.3 

Superheat 

I 

VL  + 

n= 

18 

-  5.96    = 

2.185 

PE  — 

Pq  =  49.1 

Superheat 

II 

YM  + 

Vy     = 

l.°> 

-4-6.81    = 

1.91 

PM  — 

Pq  =  58.3 

Superheat 

III 

V,+ 

F,= 

13 

-7.18    = 

1.815 

PT- 

Pq  =  63.3 

THE  STEAM  TABLES  245 

Practically,  high  efficiency  means  fuel  saving  and  high  capacity 
means  economy  in  the  first  cost  of  the  engine.  The  general  incom- 
patibility of  the  two  affords  a  fundamental  commercial  problem  in 
steam  engine  design,  it  being  the  function  of  the  engineer  to  estab- 
lish a  compromise. 

419.  The  Ideal  Steam  Engine.     No  engine  using  saturated  steam  can  develop 
an  efficiency  greater  than  that  of  the  Clausius  cycle,  the  attainable  temperature 
limits  in  present  practice  being  between  100°  and  400°  F.,  or,  for  non-condensing 
engines,  between  212°  F.  and  400°  F.     The  steam  engine  is  inherently  a  wasteful 
machine ;  the  wastes  of  practice,  not  thus  far  considered  in  dealing  with  the  ideal 
cycle,  are  treated  with  in  the  succeeding  chapter. 

THE  STEAM  TABLES 

420.  Saturated  Steam.     The  table  on  pages  247,  248  is  abridged  from  Marks' 
and  Davis'  Tables  and  Diagrams  (18).     In  computing  these,  the  absolute  zero 
was  taken  at  —  459.64°  F. ;  the  values  of  h  and  nw  were  obtained  from  the  experi- 
ments of  Barnes  and  Dietrici  (68)  on  the  specific  heat  of  water;  the  mechanical 
equivalent  of  heat  was  taken  at  777.52  ;  the  pressure-temperature  relation  as  found 
by  Holborn  and  Henning  (Art.  360)  ;  the  thermal  unit  is  the  "mean  B.  t.  u."  (see 
footnote,  Art.  23)  ;  the  value  of  H  is  as  in  Art.  388 ;  and  the  specific  volumes 
were  computed  as  in  Art.  368.     The  symbols  have  the  following  significance:  — 

P  =  pressure  in  pounds  per  square  inch,  absolute ; 
T =  temperature  Fahrenheit; 
V  —  volume  of  one  pound,  cubic  feet; 
h  =  heat  in  the  liquid  above  32°  F.,  B.  t.  u. ; 
H  =  total  heat  above  32°  F.,  B.  t.  u. ; 
L  —  heat  of  vaporization  =  H  —  A,  B.  t.  u. ; 

r  =  disgregation  work  of  vaporization  =  L  —  e  (Art.  359),  B.  t.  u. ; 
nw  =  entropy  of  the  liquid  at  the  boiling  point,  above  32°  F. ; 

ne  =  entropy  of  vaporization  =  — ; 

w.  =  total  entropy  of  the  dry  vapor  =  n   -f  ne. 

421.  Superheated  Steam.     The   computations  of   Art.  417  may  suggest  the 
amount  of  labor  involved  in  solving  problems  involving  superheated  steam.     This 
is  largely  due  to  the  fact  that  the  specific  heat  of  superheated  steam  is  variable. 
Figure  177,  representing  Thomas'  experiments,  may  be  employed  for  calculations 
which  do  not  include  volumes;  and  volumes  may  be  in  some  cases  dealt  with  by 
the  Linde  formula  (Art.  363).     The  most  convenient  procedure  is  to  use  a  table, 
such  as  that  of  Heck  (71),  or  of  Marks  and  Davis,  in  the  work  already  referred  to. 
On  the  following  page  is  an  extract  from  the  latter  table.      The  values  of  k  used 
are  the  result  of  a  harmonization  of  the  determinations  of  Knoblauch  and  Jakob 
(Art.  384)  and  Holborn  and   Henning  (69)  and  other  data   (70).     They  differ 
somewhat  from  those  given  in  Fig.  170.     The  total  heat  values  are  obtained  by 


246 


APPLIED  THERMODYNAMICS 


adding  the  values  of  k(T  -  t)  over  successive  short  intervals  of  temperature  to 
the  total  heat  at  saturation  ;  the  entropy  is  computed  in  a  corresponding  manner. 
The  specific  volumes  are  from  the  Linde  formula. 


PROPERTIES  OF  SUPERHEATED  STEAM 


SUPERHEAT,  °F 

40 

90 

200 

300 

400 

500 

6OO 

Absolute  Pressure 

t  =  141.7 

191.7 

301.7 

401.7 

501.7 

601.7 

701.7 

Lbs.  per  Square  Inch 

V  =  357.8 

387.9 

453.7 

513.4 

573.1 

632.7 

692.4 

1 

11=  1122.6 

1145.3 

1195.6 

1241.5 

1287.6 

1334.1 

1381.0 

n  =  2.0069 

2.0434 

2.1145 

2.1701 

2.2218 

2.2679 

2.4100 

t  =  166.1 

216.1 

326.1 

426.1 

526.1 

626.1 

726.1 

V  =  186.1 

204-2 

234.2 

264.1 

293.9 

323.8 

353.6 

2 

77=1133.2 

1156.1 

1206.4 

1252.4 

1298.6 

1345.2 

1392.2 

n  =  1.9486 

1.9836 

2.0529 

2.1071 

2.1586 

2.2044 

2.2459 

t  =  280.1 

330.1 

440.1 

540.1 

640.1 

740.1 

840.1 

V  =  17.35 

18.61 

21.32 

23.77 

26.20 

28.61 

31.01 

25 

77=  1179.6 

1203.4 

1255.6 

1302.8 

1350.1 

1397.5 

1445.4 

n  =  1.7402 

1.7712 

1.8330 

1.8827 

1.9277 

1.9688 

2.0078 

t  =  367.8 

417.8 

527.8 

627.8 

727.8 

827.8 

927.8 

V  =  4.72 

5.07 

5.80 

6.44 

7.07 

7.69 

8.31 

100 

77=  1208.4 

1234.6 

1289.4 

1337.8 

1385.9 

1434.1 

1482.5 

n  =  1.6294 

1.6600 

1.7188 

1.7656 

1.8079 

1.8468 

1.8829 

t  =  393.1 

443.1 

553.1 

653.1 

753.1 

853.1 

953.1 

F=3.44 

3.70 

4.24 

4.71 

5.16 

5.61 

6.06 

140 

77=1215.8 

1242.8 

1298.2 

1346.9 

1395.4 

1443.8 

1492.4 

n  =  1.6031 

1.6338 

1.6916 

1.7376 

1.7792 

1.8177 

1.8533 

t  =  398.5 

448.5 

558.5 

658.5 

758.5 

858.5 

958.5 

V  =  3.22 

3.46 

3.97 

4.41 

4.84 

5.25 

5.67 

150 

77=1217.3 

1244.4 

1300.0 

1348.8 

1397.4 

1445.9 

1494.6 

n  =  1.5978 

1.6286 

1.6862 

1.7320 

1.7735 

1.8118 

1.8474 

t  =  temperature  Fahrenheit ;   V  =  specific  volume  ;  77  =  total  heat  above  32°  F. ; 
w  =  entropy  above  32°  F. 

(Condensed  from  Steam  Tables  and  Diagrams,  by  Marks  and  Davis,  with  the  per- 
mission of  the  publishers,  Messrs.  Longmans,  Green,  &  Co.) 


THEORY  OF   VAPORS 


247 


PROPERTIES  OF   DRY  SATURATED   STEAM 

(Condensed  from  Steam  Tables  and  Diagrams,  by  Marks  and  Davis,  with  the  permis- 
sion of  the  publishers,  Messrs.  Longmans,  Green,  &  Co.) 


p 

T 

V 

h 

£ 

If 

r 

««, 

«« 

«* 

1 

101.83 

333.0 

69.8 

1034.6 

1104.4 

972.9 

0.1327 

1.8427 

1.9754 

2 

126.15 

173.5 

94.0 

1021.0 

1115.0 

956.7 

0.1749 

1.7431 

1.9180 

8 

141.52 

118.5 

109.4 

1012.3 

1121.6 

946.4 

0.2008 

1.6840 

1.8848 

4 

153.01 

90.5 

120.9 

1005.7 

1126.5 

938.6 

0.2198 

1.6416 

1.8614 

5 

162.28 

73.33 

130.1 

1000.3 

1130.5 

932.4 

0.2348 

1.6084 

1.8432 

6 

170.06 

61.89 

137.9 

995.8 

1133.7 

927.0 

0.2471 

1.5814 

1.8285 

7 

176.85 

53.56 

144.7 

991.8 

1136.5 

922.4 

0.2579 

1.5582 

1.8161 

8 

182.86 

47.27 

150.8 

988.2 

1139.0 

918.2 

0.2673 

1.5380 

1.8053 

9 

188.27 

42.36 

156.2 

985.0 

1141.1. 

914.4 

0.2756 

.5202 

1.7958 

10 

193.22 

38.38 

161.1 

982.0 

1143.1 

910.9 

0.2832 

.5042 

1.7874 

11 

197.75 

35.10 

165.7 

979.2 

1144.9 

907.8 

0.2902 

.4895 

1.7797 

12 

201.96 

32.36 

169.9 

976.6 

1146.5 

904.8 

0.2967 

.4760 

1.7727 

13 

205.87 

30.03 

173.8 

974.2 

1148.0 

902.0 

0.3025 

.4639 

1.7664 

14 

209.55 

28.02 

177.5 

971.9 

1149.4 

899.3 

0.3081 

.4523 

1.7604 

15 

213.0 

26.27 

181.0 

969.7 

1150.7 

896.8 

0.3133 

.4416 

1.7549 

16 

216.3 

24.79 

184.4 

967.6 

1152.0 

894.4 

0.3183 

.4311 

1.7494 

17 

219.4 

23.38 

187.5 

965.6 

1153.1 

892.1 

0.3229 

.4215 

1.7444 

18 

222.4 

22.16 

190.5 

963.7 

1154.2 

889.9 

0.3273 

.4127 

1.7400 

19 

225.2 

21.07 

193.4 

961.8 

1155.2 

887.8 

0.3315 

.4045 

1.7360 

20 

228.0 

20.08 

196.1 

960.0 

1156.2 

885.8 

0.3355 

.3965 

1.7320 

21 

230.6 

19.18 

198.8 

958.3 

1157.1 

883.9 

0.3393 

.3887 

1.7280 

22 

233.1 

18.37 

201.3 

956.7 

1158.0 

882.0 

0.3430 

.3811 

1.7241 

23 

235.5 

17.62 

203.8 

955.1 

1158.8 

880.2 

0.3465 

.3739 

1.7204 

24 

237.8 

16.93 

206.1 

953.5 

1159.6 

878.5 

0.3499 

.3670 

1.7169 

25 

240.1 

16.30 

208.4 

952.0 

1160.4 

876.8 

0.3532 

1.3604 

1.7136 

26 

242.2 

15.72 

210.6 

950.6 

1161.2 

875.1 

0.3564 

1  .3542 

1.7106 

27 

244.4 

15.18 

212.7 

949.2 

1161.9 

873.5 

0.3594 

1.3483 

1.7077 

28 

246.4 

14.67 

214.8 

947.8 

1162.6 

872.0 

0.3623 

1.3425 

1.7048 

29 

248.4 

14.19 

216.8 

946.4 

1163.2 

870.5 

0.3652 

1.3367 

1.7019 

30 

250.3 

13.74 

218.8 

945.1 

1163.9 

869.0 

0.3680 

1.3311 

1.6991 

31 

252.2 

13.32 

220.7 

943.8 

1164.5 

867.6 

0.3707 

1.3257 

1.6964 

32 

254.1 

12.93 

222.6 

942.5 

1165.1 

866.2 

0.3733 

1.3205 

1.6938 

33 

255.8 

12.57 

224.4 

941.3 

1165.7 

864.8 

0.3759 

1.3155 

1.6914 

34 

257.6 

12.22 

226.2 

940.1 

1166.3 

863.4 

0.3784 

1.3107 

1.6891 

35 

259.3 

11.89 

227.9 

938.9 

1166.8 

862.1 

0.3808 

1.3060 

1.6868 

36 

261.0 

11.58 

229.6 

937.7 

1167.3 

860.8 

0.3832 

1.3014 

1.6846 

37 

262.6 

11.29 

231.3 

936.6 

1167.8 

859.5 

0.3855 

1.2969 

1.6824 

38 

264.2 

11.01 

232.9 

935.5 

1168.4 

858.3 

0.3877 

1.2925 

1.6802 

39 

265.8 

10.74 

234.5 

934.4 

1168.9 

857.1 

0.3899 

1.2882 

1.6781 

40 

267.3 

10.49 

236.1 

933.3 

1169.4 

855.9 

0.3920 

1.2841 

1.6761 

41 

268.7 

10.25 

237.6 

932.2 

1169.8 

854.7 

0.3941 

1.2800 

1  6741 

42 

270.2 

10.02 

239.1 

931.2 

1170.3 

853.6 

0.3962 

1.2759 

1.6721 

43 

271.7 

9.80 

240.5 

930.2 

1170.7 

852.4 

0.3982 

1.2720 

1.6702 

44 

273.1 

9.59 

242.0 

929.2 

1171.2 

851.3 

0.4002 

1.2681 

1.6683 

45 

274.5 

9.39 

243.4 

928.2 

1171.6 

850.3 

0.4021 

1  .2644 

1.6665 

46 

275.8 

9.20 

244.8 

927.2 

1172.0 

849.2 

0.4040 

1.2607 

1.6647 

47 

277.2 

9.02 

246.1 

923.3 

1172.4 

848.1 

0.4059 

1.2571 

1.6630 

48 

278.5 

8.84 

247.5 

925.3 

1172.8 

847.1 

0.4077 

1.2536 

1.6613 

49 

279.8 

8.67 

248.8 

924.4 

1173.2 

846.1 

0.4095 

1.2502 

1.6597 

50 

281.0 

8.51 

250.1 

923.5 

1173.6 

845.0 

0.4113 

1.2468 

1.6581 

248 


APPLIED  THERMODYNAMICS 


PROPERTIES   OF   DRY   SATURATED   STEAM  —  CONTINUED 

(Condensed  from  Steam  Tables  and  Diagrams,  by  Marks  and  Davis,  with  the  permis- 
sion of  the  publishers,  Messrs.  Longmans,  Green,  &  Co.) 


p 

T 

V 

h 

L 

II 

r 

nw 

«« 

KS 

51 

282.3 

8.35 

251.4 

922.6 

1174.0 

844.0 

0.4130 

1.2435 

1.6565 

52 

283.5 

8.20 

252.6 

921.7 

1174.3 

843.1 

0.4147 

1.2402 

1.6549 

53 

284.7 

8.05 

253.9 

920.8 

1174.7 

842.1 

0.4164 

1.2370 

1.6534 

54 

285.9 

7.91 

255.1 

919.9 

1175.0 

841.1 

0.4180 

1.2339 

1.6519 

55 

287.1 

7.78 

256.3 

919.0 

1175.4 

840.2 

0.4196 

1.2309 

1.6505 

56 

288.2 

7.65 

257.5 

918.2 

1175.7 

839.3 

0.4212 

1.2278 

1.6490 

57 

289.4 

7.52 

258.7 

917.4 

1176.0 

838.3 

0.4227 

1.2248 

1.6475 

58 

290.5 

7.40 

259.8 

916.5 

1176.4 

837.4 

0.4242 

1.2218 

1.6460 

59 

291.6 

7.28 

261.0 

915.7 

1176.7 

836.5 

0.4257 

1.2189 

1.6446 

60 

292.7 

7.17 

262.1 

914.9 

1177.0 

835.6 

0.4272 

1.2160 

1.6432 

61 

293.8 

7.06 

263.2 

914.1 

1177.3 

834.8 

0.4287 

1.2132 

1.6419 

62 

294.9 

6.95 

264.3 

913.3 

1177.6 

833.9 

0.4302 

1.2104 

1.6406 

63 

295.9 

6.85 

265.4 

912.5 

1177.9 

833.1 

0.4316 

1.2077 

1.6393 

64 

297.0 

6.75 

266.4 

911.8 

1178.2 

832.2 

0.4330 

1.2050 

1.6380 

65 

298.0 

6.65 

267.5 

911.0 

1178.5 

831.4 

0.4344 

1.2034 

1.6368 

66 

299.0 

6.56 

268.5 

910.2 

1178.8 

830.5 

0.4358 

1.2007 

1.6355 

67 

300.0 

6.47 

269.6 

909.5 

1179.0 

829.7 

0.4371 

1.1972 

1.6343 

68 

301.0 

6.38 

270.6 

908.7 

1179.3 

828.9 

0.4385 

1.1946 

1.6331 

69 

302.0 

6.29 

271.6 

908.0 

1179.6 

828.1 

0.4398 

1.1921 

1.6319 

70 

302.9 

6.20 

272.6 

907.2 

1179.8 

827.3 

0.4411 

1.1896 

1.6307 

71 

303.9 

6.12 

273.6 

906.5 

1180.1 

826.5 

0.4424 

1.1872 

1.6296 

72 

304.8 

6.04 

274.5 

905.8 

1180.4 

825.8 

0.4437 

1.1848 

1.6285 

73 

305.8 

5.96 

275.5 

905.1 

1180.6 

825.0 

0.4449 

1.1825 

1.6274 

74 

306.7 

5.89 

276.5 

904.4 

1180.9 

824.2 

0.4462 

1.1801 

1.6263 

75 

307.6 

5.81 

277.4 

903.7 

1181.1 

823.5 

0.4474 

1.1778 

1.6252 

80 

312.0 

5.47 

282.0 

900.3 

1182.3 

819.8 

0.4535 

1.1665 

1.6200 

85 

316.3 

5.16 

286.3 

897.1 

1183.4 

816.3 

0.4590 

1.1561 

1.6151 

90 

320.3 

4.89 

290.5 

893.9 

1184.4 

813.0 

0.4644 

1.1461 

1.6105 

95 

324.1 

4.65 

294.5 

890.9 

1185.4 

809.7 

0.4694 

1.1367 

1.6061 

100 

327.8 

4.429 

298.3 

888.0 

1186.3 

806.6 

0.4743 

1.1277 

1.6020 

105 

331.4 

4.230 

302.0 

885.2 

1187.2 

803.6 

0.4789 

1.1191 

1.5980 

110 

334.8 

4.047 

305.5 

882.5 

1188.0 

800.7 

0.4834 

1.1108 

1.5942 

115 

338.1 

3.880 

309.0 

879.8 

1188.8 

797.9 

0.4877 

1.1030 

1.5907 

120 

341.3 

3.726 

312.3 

877.2 

1189.6 

795.2 

0.4919 

1.0954 

1.5873 

125 

344.4 

3.583 

315.5 

874.7 

1190.3 

792.6 

0.4959 

1.0880 

1.5839 

130 

347.4 

3.452 

318.6 

872.3 

1191.0 

790.0 

0.4998 

1.0809 

1.5807 

140 

353.1 

3.219 

324.6 

867.6 

1192.2 

785.0 

0.5072 

1.0675 

1.5747 

150 

358.5 

3.012 

330.2 

863.2 

1193.4 

780.4 

0.5142 

1.0550 

1.5692 

160 

363.6 

2.834 

335.6 

858.8 

1194.5 

775.8 

0.5208 

1.0431 

1.5639 

170 

368.5 

2.675 

340.7 

854.7 

1195.4 

771.5 

0.5269 

1.0321 

1.5590 

180 

373.1 

2.533 

345.6 

850.8 

1196.4 

767.4 

0.5328 

1.0215 

1.5543 

190 

377.6 

2.406 

350.4 

846.9 

1197.3 

763.4 

0.5384 

1.0114 

1.5498 

200 

381.9 

2.290 

354.9 

843.2 

1198.1 

759.5 

0.5437 

1.0019 

1.5456 

210 

386.0 

2.187 

359.2 

839.6 

1198.8 

755.8 

0.5488 

0.9928 

1.5416 

220 

389.9 

2.091 

363.4 

836.2 

1199.6 

752.3 

0.5538 

0.9841 

1.5379 

230 

393.8 

2.004 

367.5 

832.8 

1200.2 

748.8 

0.5586 

0.9758 

1.5344 

240 

397.4 

1.924 

371.4 

829.5 

1200.9 

745.4 

0.5633 

0.9676 

1.5309 

250 

401.1 

1.850 

375.2 

82  J.3 

1201.5 

742.0 

0.5676 

0.9600 

1.5276 

THEORY  OF   VAPORS  249 

(1)  Phil.  Trans.,  1854,  CXLIV,  360.  (2)  Phil.  Trans.,  1854,  336;  1862,  579. 
(3)  Theorie  Mecanique  de  la  Chaleur,  2d  ed.,  I,  195.  (4)  Wood,  Thermodynamics, 
1905,  396.  (5)  Wiedemann,  Ann.  der  Phys.  und  Chem.,  1880,  Vol.  IX.  (6)  Technical 
Thermodynamics  (Klein),  1907,  II,  215.  (7)  Mitteilungen  uber  Forschungsarbeiten, 
etc.,  21 ;  33.  (8)  Peabody,  Steam  Tables,  1908,  9 ;  Marks  and  Davis,  Tables  and 
Diagrams,  1909,  88 ;  Phil.  Trans.,  199  A  (1902),  149-263.  (9)  The  Steam  Engine, 
1897,  601.  (10)  Op.  cit.,  II,  App.  XXX.  (11)  The  Richards  Steam  Engine  Indica- 
tor, by  Charles  T.  Porter.  (12)  Trans.  A.  S.  M.  E.,  XI.  (13)  Dubois  ed.,  II,  ii,  1884. 
(14)  Peabody,  op.  cit.  (15)  Trans.  A  S.  M.  E.,  XII,  590.  (16)  Ann.  der  Physik,  4, 
26,  1908,  833.  (17)  Trans.  A.  S.  M.  E.,  XXX,  1419-1432.  (18)  Tables  and  Diagrams 
of  The  Thermal  Properties  of  Saturated  and  Superheated  Steam,  1909.  (19)  Zeits. 
fur  Instrumentenkunde,  XIII,  329.  (20)  Wissenschaftliche  Abhandlungen,  HI,  71. 

(21)  Sitzungsberichte  K.  A.  W.  in  Wien,  Math.-natur  Klasse,  CVII,  II,  Oct.,  1899. 

(22)  Loc.  cit.,  note  (7).     (23)   Op.  cit.,  417.     (24)   Comptes  Bendus,  LXII,  56  ;  Bull, 
de  la  Soc.  Industr.  de  Mulhouse,  CXXXIII,  129.     (25)  Boulvin's  method :  see  Berry, 
The  Temperature  Entropy  Diagram,  1906,  34.       (26)  Zeuner,  op.  cit.,  II,  207-208. 
(27)  Nichols  and  Franklin,  Elements  of  Physics,  I,  194.      (28)  Phil.  Trans.,  1869,  II, 
575.     (29)  Zeits.  Ver.  Deutsch.  Ing.,  1904,  24.     (30)  Trans.  A.  S.  M.  E.,  XXVIII,  8, 
1264.     (31)  Ann.  der  Phys.,  Leipzig,  1905,  IV,  XVIII,  739.     (32)  Zeits.  Ver.  Deutsch. 
Ing.,  Oct.  19,  1907.      (33)  Mitteil.  uber  Forschungsarb.,  XXXVI,  109.      (34)   Trans. 
A.  S.  M.  E.,  XXVIII,  10,  1695.     (35)   Tram.  A.  S.  M.  E.,  XXIX,  6,  633.     (36)  Ibid., 
XXX,  5,  533.         (37)  Ibid.,  XXX,  9,  1227.          (38)   Op.  cit.,  II,  239.          (39)  Pea- 
body,  Op.  cit.,  111.  (40)   The  Steam  Engine,  1905,  68.          (41)   Trans.  A.  S. 
M.  E.,  XXIX,  6.          (42)   Op.  cit.,  II,  Apps.  XXXIV,  XXXV,  XL,  XLIV,  XLII, 
XXXVIII.      (43)   Op.  cit.,  407  et.  seq.      (44)   Op.  cit.,  600.      (45)   Comptes  Bendus, 
CII,  1886,  1202.       (46)  Ibid.,  CXIV,  1892,  1093;  CXIII,  1891.       (47)  Zeits.  fur  die 
gesamte  Kalte-Industrie,  1895,  66-85.      (48)   Op.  cit.,  II,  App.  L.      (49)  Machines  a 
froid,  Paris,  1878.       (50)  Elleau  and  Ennis,  Jour.  Frank.  Inst.,  Mar.,  Apr.,  1898; 
Dietrici,  Zeits.  Kalte-Ind.,  1904.      (51)   Op.  cit.,  II,  App.  XLVI.      (52)  Zeits.  fur  die 
gesamte  Kalte-Industrie,  1904.     The  heavy  line  across  the  table  on  page  422  indicates 
a  break  in  continuity  between  the  two  sources  of  data.     The  same  break  is  responsible 
for  the  notable  irregularity  in  the  saturation  and  constant  dryness  curves  on  the  ammonia 
entropy  diagram,  Fig.  316.     (53)   Tables  of  the  Properties  of  Saturated  Steam  and  other 

Sapors,  1890.  (54)  See  Jacobus,  Trans.  A.  S.  M.  E.,  XII.  (55)  Jour.  Frank.  Inst., 
Dec.,  1890.  (56)  Op.  cit.,  466.  (57)  Mem.  de  Vlnstitut  de  France,  XXI,  XXVI. 
(58)  Landolt  and  Bornstein,  Physikalische-chemische  Tabellen ;  Gmelin  ;  Peabody, 
Thermodynamics,  118.  (59)  Andre"eff,  Ann.  Chem.  Pharm.,  1859.  (60)  Trans. 
A.  S.  M.  E.,  XXV,  176.  (61)  Tables,  etc.,  1890.  (62)  Comptes  Bendus,  CXIX, 
1894,  404-407.  (63)  Op.  cit.,  App.  XLVIII.  (64)  Op.  cit.,  468.  (65)  Trans. 
A.  S.  M.  E.,  XII.  (66)  Op.  cit.,  II,  App.  XXXII.  (67)  Trans.  A.  S.  M.  E.,  XXI, 
3,  406.  (68)  Wied.  Annallen,  (4),  XVI,  1905,  593-620.  (69)  Wied.  Annallen,  (4), 
XVIII,  1905,  739-756  ;  (4),  XXIII,  1907,  809-845.  (70)  Marks  and  Davis,  op.  cit.,  95. 
(71)  Trans.  A.  S.  M.  E.,  May,  1908. 

SYNOPSIS  OF   CHAPTER  XII 

The  temperature  remains  constant  during  evaporation  ;  that  of  the  liquid  is  the  same 
as  that  of  the  vapor;  increase  of  pressure  raises  the  boiling  point,  and  vice  versa  ; 
it  also  increases  the  density.  There  is  a  definite  boiling  point  for  each  pressure. 

Saturated  vapor  is  vapor  at  minimum  temperature  and  maximum  density  for  the  given 
pressure. 

Superheated  vapor  is  an  imperfect  gas,  produced  by  adding  heat  to  a  dry  saturated  vapor. 


250  APPLIED  THERMODYNAMICS 


Saturated  Steam 

PC  W  —  V} 
The  principal  effects  of  heat  are,  h  =  t  —  32,  e=     ^  *Q      ' 

77o 


r  =  L-e,  H=h  +  L 
As  p  increases,  t,  h,  e  and  H  increase,  and  r  and  L  decrease. 

H=  #212  +  0.3745(«  -  212)  -  0.00055(£  -  212)2. 

Factor  of  evaporation  =  L+^~hn)' 
9/0.4 

The  pressure  increases  more  rapidly  than  the  temperature. 
Characteristic  equation  for  steam,  pv  =  aT  —  p(l  +  ftp)  Ij^  —  a\- 
Saturated  steam  may  be  dry  or  wet.    For  wet  steam, 

h  =  AO»  L  =  xLo,  H  =  xL0  +  A0,  r  =  xr0,  e  =  xeo, 


and  the  factor  of  evaporation  is  -       .     The  volume  is  W=  V+x(  Wo-  V}  . 

970.4 

The  water  line  shows  the  volume  of  water  at  various  temperatures  ;  the  saturation  curve 
shows  the  relation  between  volume  and  temperature  of  saturated  steam.  Approxi- 
mately, pv1*  =  constant.  The  isothermal  is  a  line  of  constant  pressure. 

The  path  during  evaporation  is  (a)  along  the  water  line  (6)  across  to  the  saturation 
curve  at  constant  pressure  and  temperature.  If  superheating  occurs,  the  path  pro- 
ceeds at  constant  pressure  and  increasing  temperature  to  the  right  of  the  satura- 
tion curve. 

T 

On  the  entropy  diagram,  the  equation  of  the  water  line  is  n  =  cloge  —  .     The  distance 

between  the  water  line  and  the  saturation  curve  is  N  •=  —  .     Constant  dry  ness 

curves  divide  this  distance  in  equal  proportions.  Lines  of  constant  total  heat  may 
be  drawn.  The  specific  heat  of  steam  kept  dry  is  negative.  The  dry  ness  changes 
during  adiabatic  expansion.  The  temperature  of  inversion  is  that  temperature  at 
which  the  specific  heat  of  saturated  steam  is  zero.  The  change  of  internal  energy 
and  the  external  work  along  any  path  of  saturated  steam  may  be  represented  on  the 
entropy  diagram. 

w-  ir^lZLdT 
T     dP 

Constant  volume  lines  may  be  plotted  on  the  entropy  diagram,  permitting  of  the  trans- 
fer  of  any  point  or  path  from  the  PFto  the  TN  plane.  The  temperature  after 
expansion  at  contant  entropy  to  a  limiting  volume  can  best  be  obtained  from  the 
entropy  diagram.  « 

The  critical  temperature  is  that  temperature  at  which  the  latent  heat  becomes  zero 
(689°  F.}. 

Saturated  vapor  (dry  or  wet),  superheated  vapor,  gas  ;  physical  states  in  relation  to  the 
critical  temperature  ;  shape  of  isothermals. 

The  isodynamic  path  for  saturated  steam  touches  the  saturation  curve  at  one  point 
only. 


THEORY  OF   VAPORS  251 

Superheated  Steam 

The  specific  heat  has  been  in  doubt.     Its  value  increases  with  the  pressure,  and  varies 
with  the  temperature. 


Pl 


—  t 

Factor  of  evaporation  =     ^       p(^-t)-h0       py=  Q  649Q1  y_  22  &gl9  po<25 
PV  =  0.594  T-  0.00178  P.        J?  =  i  85.8.        y  =  ±  1.298. 
Paths  of  Vapors 

7  T         ~yT 

Adidbatic  equation  :—  =  c  \oge  —  +  ===•  Approximately,  PVn=  constant.   Values  of  n. 

t  fi  JL 

External  work  along  an  adiabatic  =  h—  H+  xr  —  XE. 
Continuously  superheated  adiabatic,  e.g., 


Adiabatic  crossing  the  saturation  curve  : 


T 

Method  of  drawing  constant  pressure  lines  on  the  entropy  diagram  :  n  =  Jcploge  — 

Method  of  drawing  lines  of  constant  total  heat. 

Use  of  the  entropy  diagram  for  graphically  solving  problems  :  dryness  after  expansion  ; 

work  done  during  expansion  ;  mixing  ;  heat  contents. 
The  Mollier  coordinates,  total  heat  and  entropy.    The  total  heat-pressure  diagram. 

Vapors  in  General 


When  the  pressure-temperature  relation  and  the  characteristic  equation  are  given,  we 
may  compute  L  for  various  temperatures,  and  the  specific  heat  of  the  vapor. 


Vapors  in  engineering,  Ammonia  :  logp  =  8.4079-?^,    —=91  —  ,    fc=0.508, 


vapor  density  =  0.597  (air  =  1),  specific  volume  of  liquid  =  0.025,  its  specific  heat 
=  1.02.  Sulphur  dioxide:  £  =  0.15438,  vapor  density  =  2.23,  specific  volume  of 
liquid  =  0.0007,  its  specific  heat  =  0.4.  PV-  26.4  T—  184  P0-22.  Pressure-tem- 
perature relation.  L  =  176  -  0.27  (t  -  32). 

Steam  Cycles 

Efficiency  =  work  done  4-  gross  heat  absorbed. 

The  Carnot  cycle  is  impracticable  :  the  steam  power  plant  operates  in  the  Clausius  cycle. 

(1 
Efficiency  of  Clausius  cycle  = — 


T-t+  XL 


252  APPLIED  THERMODYNAMICS 

Hankine  cycle    (incomplete    expansion)  —  determination    of    efficiency,   with    steam 
initially  wet  or  dry. 

Non-expansive  cycle:  efficiency  =  <*»-*  K**  -0.017), 

he-hd  +  xLb 

1433  1(^^-0.695(7-0 

Pambour  cycle  :  steam  dry  during  expansion  ;  efficiency  =  -  TJ-,  -  ; 

£/+  1  433  loge  —  -  0.695  (T-t) 

computation  of  heat  supplied  by  jacket. 

Superheated  cycle  :  efficiency  is  increased  if  the  final  dryness  is  properly  adjusted  and 

the  ratio  of  expansion  is  not  too  low. 
Numerical  comparison  of  seventeen  cycles  for  efficiency  and  capacity  :  steam  should 

be  initially  dry.     The  ratio  of  expansion  should  be  large  for  efficiency  and  small 

for  capacity. 

The  Steam  Tables 

Computation  is  from  p  (or  t)  to  t  (orp),  H,  h,  L,  -^,  F,  e,  »%  ««,,  nf,  ns. 

The  superheated  tables  give  n,  F,  //,  £,  for  various  superheats  at  various  pressures  ;  all 
values  depending  on  Hsat,  nsat,  and  kp. 

PROBLEMS 

NOTE.  Problems  not  marked  T  are  to  be  solved  without  the  use  of  the  steam 
table.  In  all  cases  where  possible,  computed  results  should  be  checked  step  by  step 
with  those  read  from  the  three  charts,  Figs.  175,  177,  185. 

Tl.  The  weight  per  cubic  foot  of  water  at  32°  F.  being  62.42,  and  at  250.3°  F., 
58.84,  compute  in  heat  units  the  external  work  done  in  heating  one  pound  of  water  at 
pressure  from  32°  to  250.3°.  (The  pressure  is  that  of  saturated  steam  at  a  temperature 
of  250.3°.) 

2.  Forp  =  100,  t  =  327.8°,  W=  4.429,  compute  h  (approximately),  H,  L,  e,  r  in 
the  order  given.  Why  do  not  the  results  agree  with  those  in  the  table  ? 

T3.  Find  the  factor  of  evaporation  for  dry  steam  at  95  Ib.  pressure,  the  feed- 
water  temperature  being  153°  F. 


T4.   Given  the  formula,  log  p  =  c  -          -  3969^5  ,  T  being  the  absolute  tempera- 

ture and  p  the  pressure  per  square  foot,  find  the  value  of  -£  for  p  =  100  Ib.  per  square 

dt 

inch,  t  =  327.8°  F.     Check  roughly  by  observing  nearest  differences  in  the  steam  table. 

T5.  What  increase  in  steam  pressure  accompanies  an  increase  in  temperature 
from  353.1°  F.  to  393.8°  F.  ?  Compare  the  percentages  of  increase  of  absolute  pressure 
and  absolute  temperature. 

T6.  Find  the  values  of  the  constants  in  the  Rankine  and  Zeuner  equations  (Art. 
363),  at  100  Ib.  pressure. 

TT.  From  Art.  363,  find  the  volume  of  dry  steam  at  240.1°  F.  in  four  ways. 
Compare  with  the  value  given  in  the  steam  table  and  explain  the  disagreement. 

8.  At  100  Ib.  pressure,  the  latent  heat  per  pound  is  888.0  ;  per  cubic  foot,  it  is 
200.3.  Find  the  specific  volume. 


PROBLEMS  253 

9.  For  the  conditions  given  in  Problem  2,  W  being  the  volume  of  dry  steam,  find 
the  five  required  thermal  properties  of  steam  95  per  cent  dry.  Find  its  volume. 

T 10.  State  the  condition  of  steam  (wet,  dry,  or  superheated)  when  (a)  p  =  100, 
t  =  327.8  ;  (&)  p  =  95,  v  =  4.0  ;  (c)  p  =  80,  t  =  360. 

11.  Determine  the  path  on  the  entropy  diagram  for  heating  from  200°  to  240°  F. 
a  fluid  the  specific  heat  of  which  is  1.00  -f  a£,  in  which  t  is  the  Fahrenheit  temperature 
and  a  =  0.0044. 

T 12.  Find  the  increases  in  entropy  during  evaporation  to  dry  steam  at  the  fol- 
lowing temperatures :  228°,  261°,  386°  F. 

T 13.  Compute  from  Art,  368  the  specific  volume  of  dry  steam  at  327.8  F.  What 
is  its  volume  if  4  per  cent  wet  ?  (See  Problem  4.) 

T 14.  Find  the  entropy,  measured  from  32°  F.,  of  steam  at  327.8°  F.,  65  per  cent 
dry,  (a)  by  direct  computation,  (6)  from  the  steam  table.  Explain  any  discrepancy. 

T  15.  Dry  steam  £t  100  Ib.  pressure  is  compressed  without  change  of  internal 
energy  until  its  pressure  is  200  Ib.  Find  its  dryness  after  compression. 

T 16.   Find  the  dryness  of  steam  at  300°  F.  if  the  total  heat  is  800  B.  t.  u. 

T  17.    Find  the  entropy  of  steam  at  130  Ib.  pressure  when  the  total  heat  is  840  B.  t.  u. 

T  18.  One  pound  of  steam  at  300°  F.,  having  a  total  heat  of  800  B.  t.  u.,  expands 
adiabatically  to  1  Ib.  pressure.  Find  its  dryness,  entropy,  and  total  heat  after  expan- 
sion. What  weight  of  steam  was  condensed  during  expansion  ? 

19.  Transfer  a  wet  steam  adiabatic  from  the  TNio  the  PV  plane,  by  the  graphi- 
cal method. 

20.  Transfer  a  constant  dryness  line  in  the  same  manner. 

21.  Sketch  on  the  TN  and  PV  planes  the  saturation  curve  and  the  water  line  in 
the  region  of  the  critical  temperature. 

T  22.  At  what  stage  of  dryness,  at  300°  F.,  is  the  internal  energy  of  steam  equal 
to  that  of  dry  steam  at  228°  F  ? 

T23.  At  what  specific  volume,  at  300°  F.,  is  the  internal  energy  of  steam  equal 
to  that  of  dry  steam  at  228°  F.  ? 

T  24.  Compute  from  the  Thomas  experiments  the  total  heat  in  steam  at  100  Ib. 
pressure  and  440°  F. 

T  25.  Find  the  factor  of  evaporation  for  steam  at  100  Ib.  pressure  and  500°  F.  from 
feed  water  at  153°  F. 

T  26.  In  Problem  18,  find  the  volume  after  expansion,  and  compare  with  the  vol- 
ume that  would  have  been  obtained  by  the  use  of  Zeuner's  exponent  (Art.  394). 
Which  result  is  to  be  preferred  ? 

T  27.  Using  the  Knoblauch  and  Jacob  values  for  the  specific  heat,  and  determin- 
ing the  initial  properties  in  at  least  five  steps,  compute  the  initial  entropy  and  total 
heat  and  the  condition  of  steam  after  adiabatic  expansion  from  P  =  100,  T=  700°  F. 
to  p  =  13.  Find  its  volume  from  the  formula  in  Art.  390.  Compare  with  the  volume 
given  by  the  equation  py"1-298  =pv1-™8.  (Assume  that  the  superheated  table  shows 
the  steam  to  be  superheated  about  55°  F.  at  the  end  of  expansion.) 

T  28.  Compute  the  dryness  of  steam  after  adiabatic  expansion  from  P  —  140, 
T  =  753.1°  F.,  to  t  =  153°  F.  Find  the  change  in  volume  during  expansion. 


254  APPLIED  THERMODYNAMICS 

T  29.  Find  the  external  work  done  in  Problems  27  and  28,  along  the  expansive 
paths. 

T  30.   At  what  temperature  is  the  total  heat  in  steam  at  100  Ib.  pressure  1200  B.  t.  u.  ? 
31.  Find  the  efficiency  of  the  Carnot  cycle  between  341.3°  F.  and  101.83°  F. 

T  32.  Find  the  efficiency  of  the  Clausius  cycle,  using  initially  dry  steam  between 
the  same  temperature  limits. 

T  33.    In  Problem  32,  find  the  efficiency  if  the  steam  is  initially  60  per  cent  dry. 

T  34.  In  Problem  32,  find  the  efficiency  if  expansion  terminates  when  the  volume 
is  12  cu.  ft.  (Rankine  cycle). 

T35.   In  Problem  32,  find  the  efficiency  if  there  is  no  expansion. 
T  36.   Find  the  efficiency  of  the  Pambour  cycle  between  the  temperature  limits 
given  in  Problem  31.    How  much  heat  is  supplied  by  the  jacket  ? 

T37.  Find  the  efficiency  of  this  Pambour  cycle  if  expansion  terminates  when  the 
volume  is  12  cu.  ft. 

T  38.  Steam  initially  at  140  Ib.  pressure  and  443.1°  F.  is  worked  (a)  in  the  Clau- 
sius cycle,  (&)  in  the  Rankine  cycle,  with  the  same  ratio  of  expansion  as  in  Problem 
37.  Find  the  efficiency  in  each  case,  the  lower  temperature  being  101.83°  F.  Find  the 
efficiency  of  the  Rankine  cycle  in  which  the  maximum  volume  is  5  cu.  ft.  (See  foot- 
note, Case  VIII,  Art.  417.) 

T  39.  At  what  per  cent  of  dryness  is  the  volume  of  steam  at  100  Ib.  pressure 
3  cu.  ft.  ? 

T40.  Steam  at  100  Ib.  pressure  is  superheated  so  that  adiabatic  expansion  to 
261°  F.  will  make  it  just  dry.  Find  its  condition  if  adiabatic  expansion  is  then  carried 
on  to  213°  F.  Find  the  external  work  done  during  the  whole  expansion. 

T41.  Steam  passes  adiabatically  through  an  orifice,  the  pressure  falling  from  140 
to  100  Ib.  When  the  inlet  temperature  of  the  steam  is  500°  F.,  its  outlet  temperature 
is  494°  F. ;  and  when  the  inlet  temperature  is  600°  F.,  the  outlet  temperature  is  595°  F. 
The  mean  value  of  the  specific  heat  at  140  Ib.  pressure  between  600°  F.  and  500°  F.  is 
0.498.  Find  the  mean  value  at  100  Ib.  pressure  between  595°  F.  and  494°  F.  How 
does  this  value  agree  with  that  found  by  Knoblauch  and  Jacob  ? 

T42.  Find  from  Problem  41  and  Fig.  171  the  total  heat  in  saturated  steam  at  140 
Ib.  pressure,  in  two  ways,  that  at  100  Ib.  pressure  being  1186.3. 

T43.   Plot  on  a  total  heat-pressure  diagram  the  saturation  curve,  the  constant 
dryness  curve  for  x  =  0.85,  the  constant  temperature  curve  for  T  =  500°  F.,  and  a 
constant  volume  curve  for  V  =  13,  passing  through  both  the  wet  and  the  superheated 
regions.    Use  a  vertical  pressure  scale  of  1  in.  =  20  Ib.,  and  a  horizontal  heat  scale  of  . 
1  in.  =  20  B.  t.  u. 

44.  Compute  the  temperature  of   inversion  of   ammonia,   given  the  equation, 
L  =  555.5  -  0.613  T°  F.,  the  specific  heat  of  the  liquid  being  1.0.     What  is  the  result 
if  L  =  555.5  -  0.613  T-  0.000219  T2  (Art.  401)? 

45.  Compute  the  pressure  of  the  saturated  vapor  of  sulphur  dioxide  at  60°  F.  (Art. 
404).     (Compare  Table,  page  424.) 

T46.    Compare  the  capacities  of  the  cycles  in  Problems  31-37,  as  in  Art.  418. 

47.  Sketch  the  water  line,  the  saturation  curve,  an  adiabatic  for  saturated  steam, 
and  a  constant  dryness  line  on  the  PT  plane. 


PROBLEMS  255 

T48.  A  10-gal.  vessel  contains  0.1  Ib.  of  water  and  0.7  Ib.  of  dry  steam.  What  is 
the  pressure  ? 

T49.  A  cylinder  contains  0.25  Ib.  of  wet  steam  at  58  Ib.  pressure,  the  volume 
being  1.3  cu.  ft.  What  is  the  quality  of  the  steam  ? 

T  50.   What  is  the  internal  energy  of  the  substance  in  the  cylinder  in  Problem  49  ? 

T  51.  Steam  at  140  Ib.  pressure,  superheated  400°  F.,  expands  adiabatically  until 
its  pressure  is  5  Ib.  Find  its  final  quality  and  the  ratio  of  expansion. 

T  52.  The  same  steam  expands  adiabatically  until  its  dryness  is  0.98.  Find  its 
pressure. 

T  53.  *  The  same  steam  expands  adiabatically  until  its  specific  volume  is  50.  Find 
its  pressure  and  quality. 

T54.  Steam  at  200  Ib.  pressure,  94  per  cent  dry,  is  throttled  as  in  Art.  387.  At 
what  pressure  must  the  throttle  valve  be  set  to  discharge  dry  saturated  steam  ? 

T  55.  Steam  is  throttled  from  200  Ib.  pressure  to  15  Ib.  pressure,  its  temperature 
becoming  235.5°  F.  What  was  its  initial  quality  ?  (Use  Fig.  175.) 

56.  Represent  on  the  entropy  diagram  the  factor  of  evaporation  of  superheated 
steam. 

57.  Check  by  accurate  computations  all  the  values  given  in  the  saturated  steam 
table  for  t  =  180°  F.,  using  —  459.64°  F.  for  the  absolute  zero,  14.696  Ib.  per  square 
inch  for  the  standard  atmosphere,  777.52  for  the  mechanical  equivalent  of  heat,  and 
0.017  as  the  specific  volume  of  water.     Use  Thiesen's  formula  for  the  pressure  : 


(t  +  459.6)  log  -£—  =  5.409  (t  -  212)-  8.71  x  10-10[(689-  «)4  -  477*]; 

t  being  the  Fahrenheit  temperature  and  p  the  pressure  in  pounds  per  square  inch.  Use 
the  Knoblauch,  Linde  and  Klebe  formula  for  the  volume  and  the  Davis  formula  for 
the  total  heat.  Compute  the  entropy  and  heat  of  the  liquid  in  eight  steps,  using  the 
following  values  for  the  specific  heat  of  the  liquid : 

at  40°,    1.0045;  at  120°,  0.9974  ; 

at  60°,    0.9991  ;  at  140°,  0.9987  ; 

at  80°,    0.997  ;  at  160°,  1.0002  ; 

at  100°,  0.99675  ;  at  180°,  1.0020. 

Explain  the  reasons  for  any  discrepancies. 

T  58.  Check  the  properties  given  in  the  superheated  steam  table  for  P  =  25  with 
200°  of  superheat,  using  Knoblauch  values  for  the  specific  heat,  in  at  least  three  steps, 
and  using  the  Knoblauch,  Linde  and  Klebe  formula  for  the  volume.  Explain  any 
discrepancies. 

59.    Represent  on  the  entropy  diagram  the  temperature  of  inversion  of  a  dry  vapor. 

*  This  is  typical  of  a  class  of  problems  the  solution  of  which  is  difficult  or  impos- 
sible without  plotting  the  properties  on  charts  like  those  of  Figs.  175,  177,  185.  Prob- 
lem 53  may  be  solved  by  a  careful  inspection  of  the  total  heat-pressure  and  Mollier 
diagrams,  with  reasonable  accuracy.  The  approximate  analytical  solution  will  be  found 
an  interesting  exercise.  We  have  no  direct  formula  for  relation  between  V  and  T, 
although  one  may  be  derived  by  combining  the  equations  of  Rankine  or  Zeuner  (Art, 
363)  with  that  in  Problem  4. 


CHAPTER   XIII 

THE   STEAM  ENGINE 
PRACTICAL  MODIFICATIONS  OF  THE  RANKINE  CYCLE 

422.  The  Steam  Engine.     Figure  186  shows  the  working  parts. 
The  piston  P  moves  in  the  cylinder  A,  communicating  its  motion 
through  the  piston  rod  72,  crosshead  (7,  and  connecting  rod  M  to  the 
disk  crank  D  on  the  shaft  £,  and  thus  to  the  belt  wheel  W.     The 
guides  on  which  the  crosshead  moves  are  indicated  by   6r,  H,  the 
frame  which  supports  the  working  parts  by  F.  °  Journal   bearings 
at  B  and  0  support  the  shaft     The  function  of  the  mechanism  is  to 
transform  the  to-and-fro  rectilinear  motion  of  the  piston  to  a  rotatory 
movement  at  the  crank.     Without  entering  into  details  at  this  point, 
it  may  be  noted  that  the  valve  V,  which  alternately  admits  of  the 
passage  of  steam  through  either  of  the  ports  X,  Y,  is  actuated  by  a 
valve  rod  /  traveling  from  a  rocker  J",  which  derives  its  motion  from 
the  eccentric  rod  N  and  the  eccentric   ~E.     In  the  end  view,  L  is  the 
opening  for  the  admission  of  steam  to  the  steam  chest  K,  Q  is  a  sim- 
ilar opening  for  the  exit  of  the  steam  (shown  also  in  the  plan),  and 
Vis  the  valve. 

423.  The  Cycle.     With  the  piston  in  the  position  shown,  and 
moving  to  the  left,  steam  is  passing  from  the  steam  chest  through  Y 
into  the  cylinder,  while  another  mass  of  steam,  which  has  expended 
its  energy,  is  passing  from  the  other  side  of  the  piston  through  the 
port  X  and  the   opening    Q  to   the  atmosphere   or  the   condenser. 
When  the  piston  shall  have  reached  its  extreme  left-hand  position, 
the  valve  will  have  moved  to  the  right,  the  port  Y  will  have  been 
cut  off  from  communication  with  K,  and  the  steam  on  the  right  of 
the  piston  will  be  passing  through  l^to  Q.     At  the  same  time  the 
port  X  will  be  cut  off  from  Q  and  placed  in  communication  with  K. 
The  piston  then  makes  a  stroke  to  the  right,  while  the  valve  moves 
to  the  left.     The  engine  shown  is  thus  double-acting. 

256 


THE  STEAM   ENGINE 


257 


258 


APPLIED  THERMODYNAMICS 


If  the  valve  moved  instantaneously  from  one  position  to  the  other 
precisely  at  the  end  of  the  stroke,  the  PV  diagram  representing 
the  changes  in  the  fluid  on  either  side  of  the  piston  would  resemble 
ebtd,  Fig.  184.  Along  eb,  the  steam  would  be  passing  from  the 
steam  chest  to  the  cylinder,  the  pressure  being  practically  constant 
because  of  the  comparatively  enormous  storage  space  in  the  boiler, 
while  the  piston  moved  outward,  doing  work.  At  6,  the  supply  of 
steam  would  cease,  while  communication  would  be  immediately 
opened  with  the  atmosphere  or  the  condenser,  causing  the  fall  of 
pressure  along  bt.  The  piston  would  then  make  its  return  stroke, 
the  steam  passing  out  of  the  cylinder  at  practically  constant  pressure 
along  td,  and  at  d  the  position  of  the  valve  would  again  be  changed, 
closing  the  exhaust  and  opening  the  supply  and  giving  the  instan- 
taneous rise  of  pressure  indicated  by  de. 

424.  Expansion.  This  has  been  shown  to  be  an  inefficient  cycle 
(Art.  417),  and  it  would  be  impossible,  for  mechanical  reasons,  to 
more  than  approximate  it  in  practice.  The  inlet  port  is  nearly 
p  „  always  closed  prior 

to  the  end  of  the 
stroke,  producing 
such  a  diagram  as 
debgq,  Fig.  184,  in 
which  the  supply  of 
steam  to  the  cylin- 
der is  less  than  the 
whole  volume  of  the 
piston  displacement, 
and  the  work  area 
under  lg  is  obtained 

FIG.  187.     Arts.  424,  425,  427,  430,  431,  436,  441,  445,  446,  448,    without   the    Supply 
449, 450, 451,  452,  454.  —  Indicator  Diagram  and  Rankiue  Cycle.        £ 


in  consequence  of  the  expansive  action  of  the  steam.     Apparently, 
then,  the  actual  steam  engine  cycle  is  that  of  Rankine*  (Art.  411). 

*  It  need  scarcely  be  said  that  the  association  of  the  steam  engine  indicator  dia- 
gram and  its  varying  quantity  of  steam  with  the  ideal  Rankine  cycle  is  open  to  objec- 
tion (Art  454).  Yet  there  are  advantages  on  the  ground  of  simplicity  in  this  method 
of  approaching  the  subject. 


WIREDRAWING  259 

But  if  we  apply  an  indicator  (Art.  484)  to  the  cylinder,  —  an  instru- 
ment for  graphically  recording  the  changes  of  pressure  and  volume 
during  the  stroke  of  the  piston,  —  we  obtain  some  such  diagram  as 
abcdes,  Fig.  187,  which  may  be  instructively  compared  with  the  cor- 
responding Rankine  cycle,  ABCDE.  The  remaining  study  of  the 
steam  engine  deals  principally  with  the  reasons  for  the  differences 
between  these  two  cycles. 

425.  Wiredrawing.     The  first  difference  to  be  considered  is  that  along  the 
lines  ab,  AB.     An  important  reason  for  the  difference  in  rolumes  at  b  and  B  will 
be  discussed  (Art.  430)  ;  we  may  at  present  note  that  the  pressures  at  a  and  b  are 
less  than  those  at  A  and  B,  and  that  the  pressure  at  b  is  less  than  that  at  a.     This 
is  due  to  the  frictional  resistance  of  steam  pipes,  valves,  and  ports,  which  causes 
the  steam  to  enter  the  cylinder  at  a  pressure  somewhat  less  than  that  in  the  boiler  ; 
and  produces  a  further  drop  of  pressure  while  the  steam  enters.     The  action  of 
the  steam  in  thus  expanding  with  considerable  velocity  through  constricted  pas- 
sages  is  described   as  "  wiredrawing."     The  average  pressure  along  ab  will  not 
exceed  0.9  of  the  boiler  pressure;  it  may  be  much  less  than  this.     A  loss  of  work 
area  ensues.     The  greater  part  of  the  loss  of  pressure  occurs  in  the  ports  and  pas- 
sages of  the  cylinder  and  steam  chest.     The  friction  of  a  suitably  designed  steam 
pipe  is  small.     The  pressure  drop  due  to  wiredrawing  or  "throttling,"  as  it  is 
sometimes  called,  is  greatly  aggravated  when  the  steam  is  initially  wet;  Clark 
found  that  it  might  be  even  tripled.     Wet  steam  may  be  produced  as  a  result  of 
priming  or  frothing  in  the  boiler,  or  of  condensation  in  the  steam  pipes.     Its  evil 
effect  in  this  as  in  other  respects  is  to  be  prevented  by  the  use  of  a  steam  separator 
near  the  engine  ;  this  automatically  separates  the  steam  and  entrained  moisture, 
and  the  water  is  then  trapped  away. 

The  mean  piston  velocity  in  the  average  steam  engine  is  about  10  ft.  per  sec- 
ond. A  high  speed  of  the  piston  as  compared  with  the  velocity  of  the  steam  might 
therefore  be  expected  to  accentuate  the  pressure  drop  along  ab.  The  speed  of  the 
piston  is,  however,  always  low  as  compared  with  that  of  the  steam,  and  there  is 
consequently  a  perceptible  impact  when  steam  is  admitted.  This  leads  to  a  rise 
of  pressure,  and  the  shape  of  the  line  ab,  so  far  as  piston  speed  may  affect  it,  is 
determined  by  the  joint  effect  of  these  two  causes. 

426.  Thermodynamics  of  Throttling.     Wiredrawing  is  a  non-reversible 
process,  in  that  expansion   proceeds,   not  against  a  sensibly   equivalent 
external  pressure,  but  against  a  lower  and   comparatively  non-resistant 
pressure.     If  the  operation  be  conducted  with  sufficient  rapidity,  and  if 
the  resisting  pressure  be  negligible,  the  external  work  done  should   be 
zero,  and   the   initial   heat   contents   should   be  equal  to  the  final   heat 
contents ;  i.e.  the  steam  expands  adiabatically  (though  not  isentropically) 
along  a  line  of  constant  total  heat  like  mr,  Fig.  161.     The  steam  is  thus 
dried  by  throttling ;  but  since  the  temperature  lias  been  reduced,  the  heat 


260 


APPLIED  THERMODYNAMICS 


has  lost  availability.     Figure  188  represents  the  case  in  which  the  steam 

remains  superheated  throughout  the  throttling  process.  A  is  the  initial 

state,  DA  and  EC  lines  of  constant  pressure, 
AB  an  adiabatic,  AF  a  line  of  constant  total 
heat,  and  C  the  final  state.  The  areas 
SHJDAG  and  SHECK,  and,  consequently, 
the  areas  JDABEH  and  GBCK,  are  equal ; 
the  temperature  at  C  is  less  than  that  at  A. 
(See  the  superheated  steam  tables  :  at  p=140, 
#=1298.2  when  £  =  553.1°  F. ;  atp=100, 
#=1298.2  when  t  is  about  548°  F.)  The 
effect  of  wiredrawing  is  thus  generally  to 

lower  the  temperature,  while  leaving  the  total  quantity  of  heat  unchanged. 

The  curve  ab,  Fig.  187,  must,  in  theory  at  least,  appear  on  the  entropy 

diagram    as   a   line  of   constant 

total  heat.     On  the  ideal  entropy 

diagram  ABODE  of  Fig.  189  we 


FIG.  188.    Art.  426.  —  Throttling 
of  Superheated  Steam. 


therefore     sketch 
drawn  "  line  ab. 


the 


wire- 


Artg  m  445>  453  _  Converted  Illdi. 
cator  Diagram  and  Rankine  Cycle. 


427.  Regulation   by  Throttling. 
On   some  of   the  cheaper   types   of 
steam  engine,  the  speed  is  controlled 
by  varying  the  extent  of  opening  of 
the  admission  pipe,  thus  producing 

a  wiredrawing  effect  throughout  the    Fm 

stroke.     It  is  obvious  that   such   a 

method  of    regulation    cannot    be 

other  than  wasteful  ;  a  better  method  is,  as  in  good  practice,  to  vary  the  point  of 

cut-off,  6,  Fig.  187. 

428.  Expansion  Curve.    The  widest  divergence  between  the  theoretical 
and  actual  diagrams  appears  along  the  expansion  lines  be,  BC,  Fig.  187. 
In  neither  shape  nor  position  do  the  two  lines  coincide.   Early  progress  in 
the  development  of  the  steam  engine  resulted  in  the  separation  of   the 
three  elements,  boiler,  cylinder,  and  condenser.     In  spite  of  this  separa- 
tion, the  cylinder  remains,  to  a  certain  extent,  a  condenser  as  well  as  a 
boiler,  alternately  condensing  and  evaporating  large  proportions  of  the 
steam  supplied,  and  producing  erratic  effects  not  only  along  the  expansion 
line,  but  at  other  portions  of  the  diagram  as  well. 

429.  Importance  of  Cylinder  Condensation.     The  theoretical  analysis  of  the  Ran- 
kine cycle  (Art.  411)  gives  efficiencies  considerably  greater  than  those  actually  attained 
in  practice.     The  reason  for  this  was  not  suggested  by  Rankine,  who  in  his  earlier 


CYLINDER  CONDENSATION  261 

writings  almost  wholly  ignored  the  fact  that  nothing  approaching  an  adiabatic 
condition  is  possible  with  stearn  contained  ill  the  conducting  cast-iron  walls  of  an 
engine  cylinder.  The  actual  action  was  pointed  out  by  Clark's  experiments  on 
locomotives  in  1855  (1) ;  and  still  more  comprehensively  by  Isherwood,  in  his 
classic  series  of  engine  trials  made  on  a  vessel  of  the  United  States  Navy  (2). 
The  further  studies  of  Loring  and  Emery  and  of  Ledoux  (3),  and,  most  of  all, 
those  conducted  under  the  direction  of  Him  (4),  served  to  point  out  the  vital 
importance  of  the  question  of  heat  transfers  within  the  cylinder.  Recent  accurate 
measurements  of  the  fluctuations  in  temperature  of  the  cylinder  walls  by  Hall, 
Callendar  and  Nicolson  (5)  and  at  the  Massachusetts  Institute  of  Technology  (6) 
have  furnished  quantitative  data. 

430.  Initial  Condensation.   When  hot  steam  enters  the  cylinder  at  or 
near  the  beginning  of  the  stroke,  it  meets  the  relatively  cold  surface  of 
the  piston  and  cylinder  head,  and  partial  liquefaction  immediately  occurs. 
As  the  piston  moves  forward,  more  of  the  cylinder  wall  is  exposed  to  the 
steam,  and  condensation  continues.     The  initial  condensation  is  by  far  the 
most  important  of  the  heat  exchanges  to  be  considered.     By  the   time 
the  point  of  cut-off  is  reached  the  steam  may  contain  from  25  to  70  per 
cent  of  water.     The   actual   weight  of  steam  supplied  by  the  boiler  is, 
therefore,  not  determined  by  the  volume  at  6,  Fig.  187 ;  it  is  practically 
from  33  to  233  per  cent  greater  than  the  amount  thus  determined.     If 
ABODE,  Fig.  187,  represents  the  ideal  cycle,  then  b  will  be  found  at  a  point 
where  Vb  =  from  0.30  VB  to  0.75  VB  (Art.  436). 

431.  Condensation  during  Expansion.     The  admission  valve  closes  at  6, 
and  the  steam  is  permitted  to   expand.     Condensation  continues   for   a 
time,  the  chilling  wall  surface   increasing.     As  expansion   proceeds   the 
pressure  of  the  steam  falls  until  its  temperature  becomes  less  than  that  of 
the  cylinder  walls,  when  an  opposite  transfer  of  heat  begins.     The  ivalls 
now  give  up  heat  to  the  steam,  drying  it,  i.e.  evaporating  a  portion  of  the 
commingled  water.     At  the  moment  when  the   direction  of  heat   trans- 
fer changes,  the  percentage  of  water  has  usually  reached  a  maximum ; 
from  that  point  onward,  it  decreases.     The  behavior  is  complicated,  how- 
ever, by  the  liquefaction  which  necessarily  accompanies  expansion,  even  if 
adiabatic  (Art.  372).    The  reevaporation  of  the  water  during  the  later  stages 
of  expansion  is  effected  by  a  withdrawal  of  heat  from  the  walls ;  these 
are  consequently  cooled,  resulting  in  the  resumption  of  proper  conditions 
for  a  repetition  of  the  whole  destructive  process  during  the  next  succeed- 
ing stroke.     Reevaporation  is  an  absorption  of  heat  by  the  fluid.     For 
maximum  efficiency,  all  heat  should  be  absorbed  at  maximum  temperature, 
as  in  the  Carnot  cycle.     The  later  in  the  stroke  that  reevaporation  occurs, 
the  lower  is  the  temperature  of  reabsorption  of  this  heat,  and  the  greater 
is  the  loss  of  efficiency.     Reevaporation  is  often  not  thermally  complete ; 


262  APPLIED  THERMODYNAMICS 

the  steam  may  iii  some  cases  be  brought  to  a  condition  of  dryness  at  the 
point  of  release,  but  in  general  the  temperature  of  the  steam  at  that  point 
is  at  least  20°  F.  below  that  of  the  cylinder  walls  (7). 

432.  Continuity  of  Action.     When  unity  of  weight  of  steam  condenses,  it  gives 
up  the  latent  heat  L ;  when  afterward  reevaporated,  it  reabsorbs  the  latent  heat  Ll ; 
meanwhile,  it  has  cooled,  losing  the  heat  h  —  hv     The  net  result  is  an  increase  of 
heat  in  the  walls  of  L  —  L^  +  h  —  hl  =  H  —  Hr  and  the  walls  would  continually  be- 
come hotter,  were  it  not  for  the  fact  that  heat  is  being  lost  by  radiation  to  the 
external  atmosphere  and  that  more  water  is  reevaporated  than  was  initially  con- 
densed ;  so  much  more  in  fact,  that  the  dryness  at  the  end  of  expansion  is  usually 
greater  than  it  would  have  been,  had  expansion  been  adiabatic. 

The  outer  portion  of  the  cylinder  walls  remains  at  practically  uniform  tem- 
perature, steadily  and  irreversibly  losing  heat  to  the  atmosphere.  The  inner  portion 
has  been  experimentally  shown  to  fluctuate  in  temperature  in  accordance  with  the 
changes  of  temperature  of  the  steam  in  contact  with  it.  The  depth  of  this  "  peri- 
odic "  portion  is  small,  and  decreases  as  the  time  of  contact  during  the  cycle 
decreases,  e.g.  in  high  speed  engines. 

433.  Influences  Affecting  Condensation.     Four  main  factors  are  related 
to  the  phenomena  of  cylinder  condensation :  they  are  (a)  the  temperature 
range,  (b)  the  size  of  the   engine,  (c)  its   speed   and   (most   important), 
(d)  the  ratio  of  volumes  during  expansion.     Of  extreme  importance,  as 
affecting  condensation  during  expansion,  is  the  condition  of  the  steam  at  the 
beginning  of  expansion.     If  this  is  wet,  either  because  of  the  delivery  of 
wet  steam  to  the  engine  or  because  of  initial  condensation  (Art.  430),  the 
condensation  during  expansion  is  greatly  increased. 

The  greater  the  range  of  pressures  (and  temperatures)  in  the  engine,  the  more 
marked  are  the  alternations  in  temperature  of  the  walls,  and  the  greater  is  the 
difference  in  temperature  between  steam  and  walls  at  the  moment  when  steam  is 
admitted  to  the  cylinder.  A  wide  range  of  working  temperatures,  although  practi- 
cally as  well  as  theoretically  desirable,  has  thus  the  disadvantage  of  lending  itself 
to  excessive  losses. 

434.  Speed.     At  infinite  speed,  there  would  be  no  time  for  the  transfer  of 
heat,  however  great  the  difference  of  temperature.     Willans  has  shown  the  per- 
centage of  water  present  at  cut-off  to  decrease  from  20.2  to  5.0  as  the  speed  in- 
creased from  122  to  401  r.  p.  m.,  the  steam  consumption  per  Ihp-hr.  concurrently 
decreasing  from  27.0  to  24.2  Ib.  (8).    In  another  test  by  Willans,  the  speed  ranged 
from  131  to  405  r.  p.  m.,  the  moisture  at  cut-off  from  29.7  to  11.7,  and  the  steam 
consumption  from  23.7  to  20.3 ;  and  in  still  another,  the  three  sets  of  figures  were 
116  to  401,  20.9  to  8.9,  and  20.0  to  17.3.     In  all  cases,  for  the  type  of  engine  under 
consideration,  increase  of  speed  decreased  the  proportion  of  moisture  and  increased 
the  economy :  but  it  should  not  be  inferred  from  this  that  high  speeds  are  neces- 
sarily or  generally  associated  with  highest  efficiency. 


EXPANSION   AND   CYLINDER   CONDENSATION       263 


435.  Size.  The  volume  of  a  cylinder  is  irD^L  -4-  4  and  its  exposed  wall  sur- 
face is  (irDL)  -+•  (nD2  -f-  2),  if  D  denotes  the  diameter  and  L  the  exposed  length. 
The  volume  increases  more  rapidly  than  the  wall  surface,  as  the  diameter  is  in- 
creased for  a  constant  length.  Since  the  lengths  of  cylinders  never  exceed  a  certain 
limit,  it  may  be  said,  generally,  that  small  engines  show  greater  amounts  of  con- 
densation, and  lower  efficiencies,  than  large  engines. 


>.  Ratio  of  Expansion.  This  may  be  defined  as  Vd  -r-  Vb,  Fig.  187  (Art.  450). 
The  greater  the  ratio  of  expansion,  the  greater  is  the  liquefaction  accompanying 
expansion.  This  would  be  true  even  if  expansion  were  adiabatic ;  with  early  cut- 
off, moreover,  the  time  during  which  the  metal  is  exposed  to  high  temperature 
steam  is  reduced,  and  its  mean  temperature  is  consequently  less.  Its  activity  as 
an  agent  for  cooling  the  steam  during  expansion  is  thus  increased.  Again,  the 
volume  of  steam  during  admission  is  more  reduced  by  early  cut-off  than  is  the  ex- 
posed cooling  surface,  since  the  latter  includes  the  two  constant  quantities,  the 
surfaces  of  the  piston  and  of  the  cylinder  head  (clearance  ignored  (Art.  450)). 
Initial  condensation  is  thus  greatly  increased  when  the  ratio  of  expansion  is  in- 
creased, as  shown  by  Isherwood ;  and,  as  has  been  shown  (Art.  433),  excessive 
initial  condensation  leads  to  excessive  condensation  during  expansion.  The 
following  shows  the  results  of  several  experiments: 


OBSERVERS 

RATIO  OF 

PER  CENT.  OF  WATER 

STEAM  CONSUMPTION, 

EXPANSION 

AT   CCT-OFF 

POUNDS  PER  IHP-HB. 

Low 

High 

Low 

High 

Low 

High 

Loring  and  Emery 

4.2 

16.8 

.  .  . 

.    .   . 

21.2 

25.1 

Willans  (9) 

4.0 

8.0 

8.9 

25.0 

20.7 

23.1 

Barrus  (10)  gives  the  following  as  average  results  from  a  large  number  of  tests 
of  Corliss  engines  at  normal  speed : 


CUT-OFF,  PER  CENT. 
OF  STROKE 

PERCENTAGE  OF 
CONDENSATION 

CUT-OFF,  PER  CENT. 
OF  STROKE 

PERCENTAGE  OF 
CONDENSATION 

2.5 

62 

25.0 

24 

5.0 

54 

30.0 

20 

10.0 

44 

40.0 

16 

15.0 

36 

45.0 

15 

20.0 

28 

In  these  three  sets  of  experiments,  it  was  found  that  the  propor- 
tion of  water  steadily  decreased  as  the  ratio  of  expansion  decreased. 
The  steam  consumption,  however,  decreased  to  a  certain  minimum  fig- 
ure, and  then  increased.  The  beneficial  effect  of  a  decrease  in  con- 
densation was  here,  as  in  general  practice,  offset  at  a  certain  stage 


264  APPLIED  THERMODYNAMICS 

by  the  therm  odynamic  loss  due  to  relatively  incomplete-  expansion, 
discussed  in  Art.  418.  The  proper  balancing  of  these  two  factors, 
to  secure  best  efficiency,  is  the  problem  of  the  engine  designer.  It 
must  be  solved  by  recourse  to  theory,  experiment,  and  the  study  of 
standard  practice.  In  American  stationary  engines,  the  ratio  of  ex- 
pansion in  simple  cylinders  is  usually  from  4  to  5. 

437.  Quantitative  Effect.  Empirical  formulas  for  cylinder  condensation  have 
been  presented  by  Marks  and  Heck,  among  others.  Marks  (11)  gives  a  curve 
of  condensation,  showing  the  proportion  of  steam  condensed  for  various  ratios  of 
expansion,  all  other  factors  being  eliminated.  A  more  satisfactory  relation  is 
established  by  Heck  (12),  whose  formula  is 


in  which  M  is  the  proportion  of  steam  condensed  at  cut-off,  N  is  the  speed  of  the 
engine  (r.  p.  m.),  s  is  the  quotient  of  the  exposed  surface  of  the  cylinder  in 
square  feet  by  its  volume  in  cubic  feet,  p  is  the  absolute  pressure  per  square  inch 
at  cut-off,  e  is  the  reciprocal  of  the  ratio  of  expansion,  and  T  is  the  temperature  at 
cut-off.  Heck  estimates  that  the  steam  consumption  of  an  engine  may  -be  com- 
puted from  its  indicator  diagram  (Art.  500)  within  10  per  cent,  by  the  application 
of  this  formula.  If  the  steam  as  delivered  from  the  boiler  is  wet,  some  modifica- 
tion is  necessary. 

438.  Reduction  of  Condensation.     Aside  from  careful  attention  to 
the  factors  already  mentioned,  the  principal  methods  of  minimizing 
cylinder  condensation  are  by  (a)  the  use  of  steam  jackets,  (£>)  super- 
heating the  steam,  and  (c)  the  employment  of  multiple  expansion. 

439.  The  Steam  Jacket.     The  thermal  interchange  represented  by  the 
expression  L  —  L^-\-li  —  7^  of  Art.  432  involves  a  continuous  supply  of 
heat  to  the  cylinder  walls,  which  may  be  expressed  from  Art.  .360  as 
0.305  (t  —  ty.     This  heat  is  removed  from  the  walls  in  one  of  three  ways  : 
because  of  (a)  water  entering  the  cylinder  with  the  steam,  (b)  liquefac- 
tion accompanying  expansion  (the  excess  of  moisture  actively  abstracting 
heat  (Art.  431)),  or  (c)  the  transfer  of  heat  from  the  cylinder  walls  to  the 
atmosphere.     To  maintain  thermal  equilibrium,  the  steam  must  supply  to 
the  walls  sufficient  heat  to  offset  these  losses.     If  we  can  reduce  any  one 
of  the  latter,  then  the  expenditure  of  heat  by  the  steam  will  be  correspond- 
ingly reduced.     The  simple  expedient  of  covering  or  "  lagging  "  the  barrel 
and  head  of  the  cylinder  is  intended  to  reduce  initial  condensation  by  de- 
creasing the  loss  of  heat  to  the  atmosphere. 

The  steam  jacket,  invented  by  Watt,  is  a  hollow  casing  enclosing  the 
cylinder  walls,  within  which  steam  is  kept  at  high  pressure.     Jackets 


STEAM   JACKETS  265 

have  often  been  mechanically  imperfect,  and  particular  difficulty  has  been 
experienced  in  keeping  then  drained  of  the  condensed  water.  In  a  few 
cases,  the  steam  has  passed  through  the  jacket  on  its  way  to  the  cylinder; 
a  bad  arrangement,  as  the  cylinder  steam  was  thus  made  wet.  It  is  usual 
practice,  with  simple  engines,  to  admit  steam  to  the  jacket  at  full  boiler 
pressure ;  and  in  some  cases  the  pressure  and  temperature  in  the  jacket 
have  exceeded  those  in  the  cylinder.  Hot-air  jackets  have  been  used,  in 
which  flue  gas  from  the  boiler,  or  highly  heated  air,  was  passed  about  the 
body  of  the  cylinder. 

440.  Arguments  for  and  against  Jackets.  The  exposed  heated  surface 
of  the  cylinder  is  increased  and  its  mean  temperature  is  raised;  the 
amount  of  heat  lost  to  the  atmosphere  is  thus  increased.  The  jacket  is  at 
one  serious  disadvantage:  its  heat  must  be  transmitted  through  the  entire 
thickness  of  the  walls ;  while  the  internal  heat  transfers  are  effected  by 
direct  contact  between  the  steam  and  the  inner  "  skin  "  of  the  walls.  The 
use  of  a  jacket  might  seem  likely  to  lead  to  excessive  heating  of  the  steam 
during  the  exhaust  stroke,  thus  raising  the  pressure  and  causing  a  resistance 
to  the  movement  of  the  piston.  The  fact  is,  however,  that  no  such  effect  is 
produced,  because  the  steam  is  dry  or  nearly  so,  and  practically  a  non-con- 
ductor of  heat,  during  the  exhaust  stroke.  Unjacketed  cylinder  walls  act 
like  heat  sponges.  The  difference  in  mean  temperature  between  walls  and 
steam  would  not  alone  account  for  excessive  condensation,  if  the  steam 
initially  were  dry.  Small  proportions  of  moisture  greatly  facilitate  the 
heat  transfers. 

The  function  of  the  jacket  is  preventive,  rather  than  remedial,  oppos- 
ing the  formation  of  moisture  early  in  the  stroke,  liquefaction  being 
transferred  from  the  cylinder  to  the  jacket,  where  its  influence  is  less 
harmful.  The  walls  are  kept  hot  at  all  times,  instead  of  being  periodi- 
cally heated  and  cooled  by  the  action  of  the  cylinder  steam.  The  steam 
in  the  jacket  does  not  expand ;  its  temperature  is  at  all  times  the  maxi- 
mum temperature  attained  in  the  cycle.  The  mean  temperature  of  the 
walls  is  thus  raised ;  it  may  even  be  equal  to  that  of  the  steam  during  ad- 
mission, instead  of  being  50°  lower,  as  was  found  by  Donkin,  with  an  un- 
jacketed  cylinder.  The  detrimental  influence  of  the  walls  is  in  all  cases 
mitigated;  the  working  fluid  in  the  cylinder  is,  on  the  whole,  gaining 
rather  than  losing  heat  during  expansion.  The  higher  mean  temperature 
of  the  walls  makes  reevaporation  begin  earlier,  and  thus  raises  the  tem- 
perature of  reception  of  the  proportion  of  total  heat  thus  supplied.  fc, 

441.  Results  of  Jacketing.  In  the  ideal  case,  the  action  of  the  jacket  may  be 
regarded  as  shown  by  the  difference  of  the  areas  dekl  and  debf,  Fig.  183.  The 
total  heat  supplied,  without  the  jacket,  is  Ideb2,  but  cylinder  condensation  makes 


266 


APPLIED  THERMODYNAMICS 


the  steam  wet  at  cut-off,  giving  the  work  area  dekl  only.  The  additional  heat 
26/3,  supplied  by  the  jacket,  gives  the  additional  work  area  kbfl,  manifestly  at 
high  efficiency.  In  this  country,  jackets  have  been  generally  employed  on  well- 
known  engines  of  high  efficiency,  particularly  on  slow  speed  pumping  engines ;  but 
their  use  is  not  common  with  standard  designs.  Slow  speed  and  extreme  expan- 
sion, which  suggest  jackets,  lead  to  excessive  bulk  and  first  cost  of  the  engine. 
With  normal  speeds  and  expansive  ratios,  the  engine  is  cheaper  and  the  necessity 
for  the  jacket  is  less.  The  use  of  the  jacket  is  to  be  determined  from  considera- 
tions of  capital  charge,  cost  of  fuel  and  load  factor,  as  well  as  of  thermodynamic 
efficiency.  These  commercial  factors  account  for  the  far  more  general  use  of  the 
jacket  in  Europe  than  in  the  United  States. 

From  7  to  12  per  cent  of  the  whole  amount  of  steam  supplied  to  the  engine 
may  be  condensed  in  the  jacket.  The  power  of  the  engine  is  invariably  in- 
creased by  a  greater  percentage  than  that  of  increase 
of  steam  consumption.  The  cylinder  saves  more  than 
the  jacket  spends,  although  in  some  cases  the  amount 
of  steam  saved  has  been  small.  The  range  of  saving 
may  be  from  2  or  3  up  to  25  per  cent.  The  in- 
creased power  of  the  engine  is  represented  by  the 
difference  between  the  areas  abodes  and  aXYCes, 
Fig.  187.  The  latter  area  approaches  much  more 
closely  the  ideal  area  ABODE.  Jacketing  pays 


TE 

is 


_     _     V12       V8          .„         _      „ 

POINT  OF  CUT-OFF 

FIG.   190.      Art.  441.  — Effect 

of  Jackets  at  Various  Ex-     best  when  the  condltlons  are  such  as  to  naturally 
pansion  Ratios. 


induce  excessive  initial  condensation.     The  diagram 
of  Fig.  190,  after  Donkin   (14),  shows  the  variation 

in  value  of  a  steam  jacket  at  varying  ratios  of  expansion  in  the  same  engine  run 

at  constant  speed  and  initial  pressure. 

442.  Use  of  Superheated  Steam.  The  thermodynamic  advantage  of 
superheating,  though  small,  is  not  to  be  ignored,  some  heat  being  taken 
in  at  a  temperature  higher  than  the  mean  temperature  of  heat  absorption ; 
the  practical  advantages  are  more  important.  By  superheating,  a  smaller 
weight  of  steam  may  be  made  to  deliver  a  given  quantity  of  heat  to  the 
cylinder.  Adequate  superheat  fills  the  "heat  sponge"  formed  by  the 
walls,  without  letting  the  steam  become  wet  in  consequence.  If  super- 
heating is  slight,  the  steam,  during  admission,  may  be  brought  down  to 
the  saturated  condition,  and  may  even  become  wet  at  cut-off,  following 
such  a  path  as  debxbU,  Fig.  183.  With  a  greater  amount  of  superheat, 
the  steam  may  remain  dry  or  even  superheated  at  cut-off,  giving  the  paths 
debzyf,  debkzA.  The  minimum  amount  of  superheat  ordinarily  necessary 
to  give  dryness  at  cut-off  seems  to  be  about  100°  F. ;  it  may  be  much 
greater.  Ripper  finds  (15)  that  about  7.5°  F.  of  superheat  are  necessary 
for  each  1  per  cent  of  wetness  at  cut  off  to  be  expected  when  working 
with  saturated  steam.  We  thus  obtain  Fig.  191,  in  which  the  increased 
work  areas  acbd,  cefb,  eghf&re  obtained  by  superheating  along  jk,  kl,  Im, 
each  path  representing  75°  of  superheat.  Taking  the  pressure  along  ag 


SUPERHEAT 


267 


as  120  lb.,  and  that  along  lib  as  1  lb.,  the  absolute  temperatures  are  800.9° 
and  561.43°,  respectively,  and  since  the  latent  heat  at  120  lb.  is  877.2 
B.  t.  u.,  the  work  gained  by  each  of  the  areas  in 
question  is 


dbJJt 


If  we  take  the  specific  heat  of  superheated 
steam,  roughly,  at  0.48,  the  heat  used  in  secur- 
ing this  additional  work  area  is  0.48  x  75  =  36 
B.  t.  u.  The  efficiency  of  superheating  is  then 
26.1 -r- 36  =  0.73,  while  that  of  the  non-super-  FIG.  191.  Art.  442.— Super- 
heated cycle  as  a  whole,  even  if  operated  at  Car-  heat  for  overcoming  initial 
not  efficiency,  cannot  exceed  239: 47 -=-800.9 =0.30. 

Great  care  should  be  taken  to  avoid  loss  of  heat  in  pipes  between  the  super- 
heater and  the  cylinder;  without  thorough 
insulation  the  fall  of  temperature  here  may 
be  so  great  as  to  considerably  increase  the 
amount  of  superheating  necessary  to  secure 
the  desired  result  in  the  cylinder. 

The  actual  path  due  to  superheating  in 
practice  is  not  quite  as  simple  as  those  sug- 
gested in  Fig.  183.  In  Fig.  192,  let  the  path 
as  heretofore  conceived  be  ABCFG.  If  there 
is  wiredrawing  during  admission,  the  pres- 
sure at  cut-off  may  be  represented  by  the 
line  HJj  and  the  path  CFQ  will  be  replaced 
by  CM)  KL  being  a  line  of  constant  total  heat  through  F.  Expansion 
then  begins  at  M  instead  of  F.  L, 


N 


FIG.  192.  Art.  442.  — Superheat 
as  affected  by  Radiation  and 
Wiredrawing. 


X550 
£500- 


|250 
=  200 
-•150 


443.  Experimental  Results  with  Super- 
heat. The  Alsace  tests  of  1892  showed,  with 
from  60°  to  80°  of  superheat,  an  average  net 
saving  of  12  per  cent,  even  when  the  coal  con- 
sumed in  the  separately  fired  superheaters 
was  considered ;  and  when  the  superheaters 
were  fired  by  waste  heat  from  the  boilers, 
the  average  saving  was  20  per  cent.  Willans 
found  a  considerable  saving  by  superheat, 
even  when  cut-off  was  at  half  stroke,  a  ratio 
of  expansion  certainly  not  unduly  favorable 
to  superheating.  As  with  jackets,  the  ad- 
vantage of  superheat  is  greatest  in  engines 
of  low  speeds  and  high  expansive  ratios.  Striking  results  have  been  obtained  by 
the  use  of  high  superheats,  ranging  from  200°  to  300°  F.  above  the  temperature 


INDICATED  HORSE  POWER 

FIG.  193.    Art.  443,  Prob.  7.  — Steam 
Economy  in  Relation  to  Superheat. 


268  APPLIED  THERMODYNAMICS 

of  saturation.  The  mechanical  design  of  the  engine  must  then  be  considerably 
modified.  Vaughan  (16)  has  reported  remarkably  large  savings  due  to  superheat- 
ing in  locomotive  practice.  Figure  193  shows  the  decreased  steam  consumption 
due  to  various  degrees  of  superheat  in  a  small  high-speed  engine. 


.VI 
\ 


444.  Superheat  KS.  High  Pressure.    In  Fig.  194,  the  work  area  CEFHD  is 
gained  as  a  result  of  superheating  along  EF.     This 
\  may  even  exceed  the  additional  heat  absorbed,  JEFK, 

on  account  of  the  reduction  of  initial  condensation. 
By  increasing  the  initial  pressure,  the  area  BLNC 
might  have  been  gained,  but  at  an  expenditure  for  heat 
of  (practically)  LMEB,  always  greater  than  the  addi- 
tional work  obtained.  With  efficient  superheaters,  a 
given  amount  of  heat  in  superheated  steam  may  be 
delivered  to  the  engine  at  the  same  cost  as  the  same 
FIG  194  Art  444  —Super-  amount  °f  heat  in  saturated  steam;  but  the  latter 
heat  vs.  High  Pressure.  gives  a  less  efficient  cycle  in  the  cylinder  than  the 
former.  High  pressure  soon  reaches  a  mechanical 

limit;  the  limit  is  not  as  quickly  reached  with  superheat,  although  minor  diffi- 
culties in  lubrication  have  been  experienced. 


445.  Actual  Expansion  Curve.  In  Fig.  187,  1>Y  represents  the 
curve  of  saturation,  bO  the  adiabatic.  The  actual  expansion  curve 
in  an  unjacketed  cylinder  using  saturated  steam  will  then  be  some 
such  line  as  be,  the  entropy  and  fraction  of  dryriess  xy  -+-  xz  first  de- 
creasing (condensation)  and  afterwards  increasing  (reevaporation) 
as  expansion  proceeds.  Expressed  exponentially,  the  value  of  n  for 
such  expansion  curve  is  less  than  that  for  the  adiabatic  or  the  curve 
of  saturation;  in  actual  practice  it  is  always  close  to  1.00,  whence 
the  equation  of  the  curve  is  P  V=  pv.  It  should  not  be  confused  with 
the  perfect  gas  isothermal ;  that  the  equation  has  the  same  form  is 
accidental.  The  curve  PV=pv  is  an  equilateral  hyperbola,  com- 
monly called  the  hyperbolic  line.  It  may  be  plotted  for  comparison 
with  expansion  lines  of  actual  indicator  diagrams  by  the  methods  of 
Arts.  92,  93. 

The  actual  expansion  line  bo  of  Fig.  187  then  appears  as  bzc, 
Fig.  189.  Heat  is  first  lost  to  the  walls  ;  the  expansion  line  then 
recrosses  the  adiabatic  (note  the  point  M,  Fig.  187),  while  re- 
evaporation  causes  heat  absorption  along  zc.  The  heat  given  up 
to  the  walls  is  bzmn ;  that  reabsorbed  equals  zcom. 


MEAN   EFFECTIVE   PRESSURE  269 

446.    Work   done   during   Expansion  :     Engine   Capacity.      From 

•jr 

Art.  95,  this  is,  for  a  hyperbolic  curve,  Fig  187,  P6F"6loge  —  c-  -     As- 

J* 

sume  admission  and  exhaust  to  occur  without  change  of  pressure  ; 
the  cycle  is  then  precisely  that  represented  by  ABODE,  excepting 
that  the  expansive  path  is  hyperbolic.  Then  the  work  done  during 
admission  is  PBVB\  the  negative  work  during  exhaust  is  PDVC\ 
and  the  net  work  of  the  cycle  is 


PB  VB  +  PB  VB  log,     ?  -  PD  Vc  =  PB  VB   \  +  log,  -e  -  PD  Vc. 
*  B  N  VBJ 

The  mean  effective  pressure  or  average  ordinate  of  the  work  area  is 
obtained  by  dividing  this  by  V&  giving 


or,  letting  — £  =  r,  it  is 

r  n 


Letting  m  stand  for  this  mean  effective  pressure,  in  pounds  per 
square  inch,  A  for  the  piston  area  in  square  inches,  L  for  the  length 
of  the  stroke  in  feet,  and  iVfor  the  revolutions  per  minute,  the  total 
average  pressure  on  the  piston  is  mA  pounds,  the  distance  which  it 
moves  per  minute  is  2  LN  feet,  and  for  a  double-acting  engine  the 
work  per  minute  is  2  mALN  foot-pounds,  or  2  mALN-r-  33,000  horse 
power.  This  is  for  an  ideal  diagram,  which  is  always  larger  than  the 
actual  diagram  abodes ;  the  ratio  of  the  latter  to  the  former  gives  the 
diagram  factor,  by  which  the  computed  value  of  m  must  be  multiplied 
to  give  actual  results. 

Diagram  factors  for  various  types  of  engine,  as  given  by  Seaton,  are  as 
follows :  — 

Expansion  engine,  with  special  valve  gear,  or  with  a  separate  cut-off  valve, 
cylinder  jacketed  .  .  .  0.90 ; 

Expansion  engine  having  large  ports  and  good  ordinary  valves,  cylinders 
jacketed  .  .  .  0.86  to  0.88 ; 

Expansion  engines  with  ordinary  valves  and  gear  as  in  general  practice,  and 
unjacketed  .  .  .  0.77  to  0.81 ; 


270  APPLIED  THERMODYNAMICS 

Compound  engines,  with  expansion  valve  on  high  pressure  cylinder,  cylinders 
jacketed,  with  large  ports,  etc.  ...  0.86  to  0.88 ; 

Compound  engines  with  ordinary  slide  valves,  cylinders  jacketed,  good  ports, 
etc.  .  .  .  0.77  to  0.81 ; 

Compound  engines  with  early  cut-off  in  both  cylinders,  without  jackets  or 
separate  expansion  valves  .  .  .  0.67  to  0.77 ; 

Fast-running  engines  of  the  type  and  design  usually  fitted  in  warships 
.  .  .  0.57  to  0.77. 

447.  Capacity  vs.  Economy.    If  we  ignore  the  influence  of  con- 
densation, the  Clausius  cycle  (Art.  409),  objectionable  as  it  is  with 
regard  to  capacity  (Art.  418),  would  be  the  cycle  of  maximum  effi- 
ciency ;  practically,  when  we  contemplate  the  excessive  condensation 
that  would  accompany  anything  like  complete  expansion,  the  cycle  of 
Rankine  is  superior.    This  statement  does  not  apply  to  the  steam  tur- 
bine (Chapter  XIV).     The  steam  engine  may  be  given  an  enormous 
range  of  capacity  by  varying  the  ratio  of  expansion ;  but  when  this 
falls  above  or  below  the  proper  limits,  economy  is  seriously  sacrificed. 
In  purchasing  engines,  the  ratio  of  expansion  at  normal  load  should 
be  set  fairly  high,  else  the  overload  capacity  will  be  reduced.     In 
marine  service,  economy  of  fuel  is  of  especial  importance,  in  order  to 
save  storage   space.     Here   expansive   ratios   may  therefore   range 
higher  than  is  common  in  stationary  practice,  where  economy  in  first 
cost  is  a  relatively  more  important  factor. 

448.  The  Exhaust  Line :  Back  Pressure.   Considering  now  the  line  de  of  Fig. 
187,  we  find  a  noticeable  loss  of  work  area  as  compared  with  that  in  the  ideal 
case.     (Line  DE  represents  the  pressure  existing  outside  the  cylinder.)     This  is 
due  to  several  causes.     The  friction al  resistance  of  the  ports  and  exhaust  pipes 
(greatly  increased  by  the  presence  of  water)  produces  a  wiredrawing  effect,  mak- 
ing the  pressure  in  the  cylinder  higher  than  that  of  the  atmosphere  or  of  the  con- 
denser.    The  presence  of  air  in  the  exhaust  passages  of  a  condensing  engine  may 
elevate  the  pressure  above  that  corresponding  to  the  temperature  of  the  steam, 
and  so  cause  undesirable  resistance  to  the  backward  movement  of  the  piston. 
This  air  may  be  present  as  the  result  of  leakage,  under  poor  operating  conditions ; 
more  or  less  air  is  always  brought  in  the  cycle  with  the  boiler  feed  and  condenser 
water.     The  effect  of  these  causes  is  to  increase  the  pressure  during  release,  even 
in  good  engines,  from  1.3  to  3.3  Ib.  above  that  ideally  obtainable. 

Reevaporation  may  be  incomplete  at  the  end  of  expansion ;  it  then  proceeds 
during  exhaust,  sometimes,  in  flagrant  cases,  being  still  incomplete  at  the  end  of 
exhaust.  The  moisture  then  present  greatly  increases  initial  condensation.  The 
evaporation  of  any  water  during  the  exhaust  stroke  seriously  cools  the  cylinder 
walls ;  but  it  also  increases  the  pressure  resisting  the  movement  of  the  piston  and 


CLEARANCE  AND   COMPRESSION  271 

thus  raises  the  mean  elevation  of  the  line  de,  Fig.  187.  In  general  good  practice, 
the  steam  is  about  dry  during  exhaust ;  or  at  least  during  the  latter  portion  of  the 
exhaust. 

449.  Effect  of  Altitude.   The  possible  capacity  of  a  non-condensing  engine  is 
obviously  increased  at  low  barometric  pressures,  on  account  of  the  lowering  of  the 
line  DE,  Fig.  187.   With  condensing  engines,  the  absolute  pressure  attained  along 
DE  depends  upon  the  proportion  of  cooling  water  supplied  and  the  effectiveness 
of  the  condensing  apparatus.     It  is  practically  independent  of  the  barometric  pres- 
sure, excepting  at  very  high  vacua ;  consequently,  the  capacity  of  the  engine  is 
unchanged  by  variations  in  the  latter.     A  slightly  decreased  amount  of  power, 
however,  will  suffice  to  drive  the  air  pump  which  delivers  the  products  of  conden- 
sation against  any  lessened  atmospheric  pressure. 

450.  Clearance.    The  line  esa  does  not  at  any  point  come  in  contact  with  the 
ideal  line  EA,  Fig.  187.     In  all  engines,  there  is  necessarily  a  small  space  left 
between  the  piston  and  the  inside  of  the  cylinder  head  at  the  end  of  the  stroke. 
This  space,  with  the  port  spaces  back  to  the  contact  surfaces  of  the  inlet  valves,  is 
filled  with  steam  throughout  the  cycle.   The  distance  ts  in  the  diagram  represents  the 
volume  of  these  "  clearance  "  spaces.     The 

expansion  line  be  is  hyperbolic  with  ref- 
erence to  the  axis  OP ;  and  by  a  simple  re- 
versal of  Art.  92  and  Art.  93,  the  approxi- 
mate location  of  this  axis  may  readijy  be 
found  from  any  actual  diagram.  In  Fig. 
195, .the  apparent  ratio  of  expansion  is 

—  •     If  the  zero  volume  line  OP  be  found, 
ab 

the  real  ratio  of  expansion,  clearance  vol- 

FD 

ume  included,  is  ~TT"     The  clearance  in 

actual   engines   varies   from    2   to    10   per    FlG' 195'    Arts.  450,  451. -Real  and 

,.     ' .  ,         ,.     ,  ,,  parent  Expansion, 

cent  of  the  piston  displacement,  the  nec- 
essary amount  depending  largely  on  the  type  of  valve  gear.  In  such  an  engine 
as  that  of  Fig.  186,  it  is  necessarily  large,  on  account  of  the  long  ports.  It  is 
proportionately  greater  in  small  engines  than  in  those  of  large  size.  It  may  be 
accurately  estimated  by  placing  the  piston  at  the  end  of  the  stroke  and  filling  the 
clearance  spaces  with  a  weighed  or  measured  amount  of  water. 

451.  Compression.   A  large  amount  of  steam  is  employed  to  fill  the  clearance 
space  at  the  beginning  of  each  stroke.     This  can  be  avoided  by  closing  the  exhaust 
valve  prior  to  the  end  of  the  stroke,  as  at  e.  Fig.  187,  the  piston  then  compressing 
the  clearance  steam  along  es,  so  that  the  pressure  is  raised  nearly  or  quite  to  that 
of  the  entering  steam.     This  compression  serves  to  bring  the  piston  gently  to  rest, 
without  shock,  at  the  end  of  the  exhaust  stroke.     If  compression  is  so  complete  as 
to  raise  the  pressure  of  the  clearance  steam  to  that  carried  in  the  supply  pipe,  no 
loss  of  steam  will  be  experienced  in  filling  clearance  spaces.     The  work  expended 


272  APPLIED  THERMODYNAMICS 

in  compression  eahg,  Fig.  195,  will  be  largely  recovered  during  the  next  forward 
stroke  by  the  expansion  of  the  clearance  steam  :  the  clearance  will  thus  have  had 
little  effect  on  the  efficiency ;  the  loss  of  capacity  efa  will  be  just  balanced  by  the 
saving  of  steam,  for  the  amount  of  steam  necessary  to  fill  the  clearance  space 
would  have  expanded  along  ae,  if  no  other  steam  had  been  present. 

Complete  compression  would,  however,  raise  the  temperature  of  the  com- 
pressed steam  so  much  above  that  of  the  cylinder  walls  that  serious  condensation 
would  occur.  This  might  be  counteracted  by  jacketing,  but  in  practice  it  is  cus- 
tomary to  terminate  compression  at  some  pressure  lower  than  that  of  the  entering 
steam.  A  certain  amount  of  unresisted  expansion  then  takes  place  during  the 
entrance  of  the  steam,  giving  a  wiredrawn  admission  line.  If  the  pressure  at  s, 
Fig.  187,  is  fixed,  it  is,  of  course,  easy  to  determine  the  point  e  at  which  the 
exhaust  valve  must  close.  Considered  as  a  method  of  warming  the  cylinder  walls 
so  as  to  prevent  initial  condensation,  compression  is  "  theoretically  less  desirable 
than  jacketing,  for  in  the  former  case  the  heat  of  the  steam,  once  transformed  to 
work,  with  accompanying  heavy  losses,  is  again  transformed  into  heat,  while  in 
the  latter,  heat  is  directly  applied."  For  mechanical  reasons,  some  compression  is 
usually  considered  necessary.  It  makes  the  engine  smooth-running  and  probably 
decreases  condensation  if  properly  limited.  Compression  must  not  be  regarded  as 
bringing  about  any  nearer  approach  to  the  Carnot  cycle.  It  is  applied  to  a  very 
small  portion  only  of  the  working  substance,  the  major  portion  of  which  is 
externally  heated  during  its  passage  through  the  steam  plant. 

452.  Valve  Action.     We  have  now  considered  most  of  the  differences  between 
the  actual  and  ideal  diagrams  of  Fig.  187.     The  rounding  of  the  corners  at  &,  and 
along  cdu,  is  due  to  sluggish  valve  action ;  valves  must  be  opened  slightly  before  the 
full  effect  of  their  opening  is  realized,  and  they  cannot  close  instantaneously.     The 
round  corner  at  e  is  due  to  the  slow  closing  of  the  exhaust  valve.     The  inclined  line 
sa  shows  the  admission  of  steam,  the  shaded  work  area  being  lost  by  the  slow  move- 
ment of  the  valve.    In  most  cases,  admission  is  made  to  occur  slightly  prior  to  the 
end  of  the  stroke,  in  order  to  avoid  this  very  effect.     If  admission  is  too  early,  a 
negative  lost  work  loop,  mno,  may  be  formed.    Important  aberrations  in  the  diagram, 
and  modifications  of  the  phenomena  of  cylinder  condensation,  may  result  from 
leakage  past  valves  or  pistons :  these  are  matters  of  operating  error,  beyond  the 
scope  of  the  present  study. 

THE  STEAM  ENGINE  CYCLE  ON  THE  ENTROPY  DIAGRAM 

453.  Cylinder  Feed  and  Cushion  Steam.     Fig.  189  has  been  left  incomplete,  for 
reasons  which  are  now  to  be  considered.     It  is  convenient  to  regard  the  working 
fluid  in  the  cylinder  as  made  up  of  two  masses,  —  the  "  cushion  steam,"  which 
alone  fills  the  compression  space  at  the  end  of  each  stroke,  and  is  constantly  present, 
and  the  "  cylinder  feed,"  which  enters  at  the  beginning  of  each  stroke,  and  leaves 
before  the  completion  of  the  next  succeeding  stroke.     In  testing  steam  engines  by 
weighing  the  discharged  and  condensed  steam,  the  cylinder  feed  is  alone  measured  ; 
it  alone  is  chargeable  as  heat  consumption ;  but  for  an  accurate  conception  of  the 
cyclical  relations  in  the  cylinder,  the  influence  of  the  cushion  steam  must  be  con- 
sidered. 


CONVERTED   DIAGRAMS 


273 


In  Fig.  196,  let  abode  be  the  PV  diagram  of  the  mixture  of  cushion  steam  and 
cylinder  feed,  and  let  gh  be  the  expansion  line  of  the  cushion  steam  if  it  alone  were 
present.  The  total  volume  vq  at  any  point  q  of  the  combined  paths  is  made  up 
of  the  cushion  steam  volume  vo  and  the 
cylinder  feed  volume,  obviously  equal  to 
oq.  If  we  wish  to  obtain  a  diagram 
showing  the  behavior  of  the  cylinder 
feed  alone,  we  must  then  deduct  from 
the  volumes  around  abcde  the  correspond- 
ing volumes  of  cushion  steam.  The  point 
p  is  then  derived  by  making  vp  —  vq  —  ro, 
and  the  point  t  by  making  rt  =  ru  —  rs. 
Proceeding  thus,  we  obtain  the  diagram 
nzjklm,  representing  the  behavior  of  the 
cylinder  feed.  Along  nz  the  diagram 
coincides  with  the  OP  axis,  indicating 
that  at  this  stage  the  cylinder  contains 
cushion  steam  only. 


V 


\ 


FIG.  196. 


Arts.  453,  457.  —  Elimination  of 
Cushion  Steam. 


454.  The  Indicator  Diagram.  Our  study  of  the  ideal  cycles  in  Chapter  XII  has 
dealt  with  representations  on  a  single  diagram  of  changes  occurring  in  a  given  mass 
of  steam  at  the  boiler,  cylinder,  and  condenser,  the  locality  of  changes  of  condition 
being  ignored.  The  energy  diagram  abcdes  of  Fig.  187  does  not  represent  the 
behavior  of  a  definite  quantity  of  steam  working  in  a  closed  cycle.  The  pressure 
and  volume  changes  of  a  varying  quantity  of  fluid  are  depicted.  During  expansion, 
along  be,  the  quantity  remains  constant ;  during  compression  along  es,  the  quantity 
is  likewise  constant,  but  different.  Along  sab  the  quantity  increases  ;  while  along 
cde  it  decreases.  The  quality  or  dryness  of  the  steam  along  es  or  be  may  be  readily 
determined  by  comparing  the  actual  volume  with  the  volume  of  the  same  weight 
of  dry  steam ;  but  no  accurate  information  as  to  quality  can  be  obtained  along  the 
admission  and  release  lines  sab  and  cde.  The  areas  under  these  lines  represent 
work  quantities,  however,  and  it  is  therefore  possible  to  draw  an  entropy  diagram 
which  shall  represent  the  corresponding  heat  expenditures.  Such  a  diagram  will 

not  give  the  thermal  history  of  any  definite 
amount  of  steam ;  it  is  a  mere  projection  of 
the  PV  diagram  on  different  coordinates. 
It  tacitly  assumes  the  indicator  diagram  to 
represent  a  reversible  cycle,  whereas  in  fact 
the  operation  of  the  steam  engine  is  neither 
cyclic  nor  reversible. 

455.  Boulvin's  Method.  In  Fig.  197, 
let  abcde  be  any  actual  indicator  diagram, 
YZ  the  pressure  temperature  curve  of 
saturated  steam,  and  QR  the  curve  of  satu- 
ration, plotted  for  the  total  quantity  of 

FIG.  197.    Art.455.-TransferfromP7     steam '  in   the    cylinder    during    expansion. 
to  NT  Diagram  (Boulvin's  Method).         The  water  line  OS  and  the  saturation  curve 


274 


APPLIED  THERMODYNAMICS 


MT  are  now  drawn  for  this  same  quantity  of  steam,  on  the  entropy  plane.  To 
transfer  any  point,  like  B,  to  the  entropy  diagram,  we  draw  BD,  DK,  EH,  KT. 
BA,  A  T,  HT,  BG,  and  GF  as  in  Art.  378.  Then  F  is  the  required  point  on  the 
temperature  entropy  diagram.  By  transferring  other  points  in  the  same  way,  we 
obtain  the  area  NVFU,  representing  a  reversible  cycle  equivalent  to  the  actual 
diagram  so  far  as  heat  quantities  are  concerned.  The  expansion  line  thus  traced 
correctly  represents  the  actual  history  of  a  definite  quantity  of  fluid ;  the  com- 
pression line  is  imaginary,  because  during  compression  a  much  less  quantity  of 
fluid  is  actually  present  than  that  assumed.  It  is  not  safe  to  make  deductions  as 
to  the  condition  of  the  substance  from  the  NT  diagram,  excepting  along  the 
expansion  curve.  For  example,  the  diagram  apparently  indicates  that  the  dryness 
is  decreasing  along  the  exhaust  line  SU ;  although  we  have  seen  (Art.  448)  that 
at  this  stage  the  dryness  is  usually  increasing  (17). 

456.  Application  in  Practice.     In  order  to  thus  plot  the  entropy  diagram,  it  is 
necessary  to  have  an  average  indicator  card  from  the  engine,  and  to  know  the 
quantity  of  steam  in  the  cylinder.     This  last  is  determined  by  weighing  the  dis- 
charged condensed  steam  during  a  definite  number  of   strokes  and  adding  the 
quantity  of  clearance  steam,  assuming  this  to  be  just  dry  at  the  beginning  of 
compression,  an  assumption  closely  substantiated  by  numerous  experiments. 

457.  Reeve's  Method.     By  a  procedure  similar  to  that  described  in  Art.  453,  an 
indicator  diagram  is  derived  from  that  originally  given,  representing  the  behav- 
ior of  the  cylinder  feed  alone,  on  the  assumption  that  the  clearance  steam  works 
adiabatically  through  the  point  e,  Fig.  196.     This  often  gives  an  entropy  diagram 
in  which  the  compression  path  passes  to  the  left  of  the  water  line,  on  account  of 
the  fact  that  the  actual  path  of  the  cushion  steam  is  not  adiabatic,  but  is  occa- 
sionally less  "  steep." 

The  Reeve  diagram  accurately  depicts  the  process  between  the  points  of  cut- 
off and  release  and  those  of  compression  and  admission  so  far  as  the  cylinder  feed 
is  concerned,  only.  For  the  rest  of  the  cycle,  the  entropy  diagram  is  rather 
unsatisfactory  as  a  method  of  depicting  the  action  in  the  steam  engine  cylinder. 


N 


FIG.   198..     Art.   458.  — Condensation 
and  Reevaporation. 

shown    for   non-expansive  cycles. 


458.  Specimen  Diagrams.  In  Fig.  198, 
the  heat  lost  along  ab  is  nearly  all  regained 
along  be ;  but  it  here  comes  back  at  reduced 
temperature,  and  consequently  with  reduced 
availability.  Figure  199  shows  the  gain  by 
high  initial  pressure  and  reduced  back  pres- 
sure. The  augmented  work  areas  befc,  cfho, 
are  gained  at  high  efficiency ;  adji  and  adlk  cost 
nothing.  The  operation  of  an  engine  at  back 
pressure,  to  permit  of  using  the  exhaust  steam 
for  heating  purposes,  results  in  such  losses  as 
adji,  adlk.  Similar  gains  and  losses  may  be 
Figure  200  shows  four  interesting  diagrams 


plotted   from  actual   indicator  cards  from  a  small  engine  operated  at  constant 


MULTIPLE   EXPANSION 


275 


speed,  initial  pressure,  load,  and  ratio  of  expansion  (18).  Diagrams  A  and  C 
were  obtained  with  saturated  steam,  B  and  D  with  superheated  steam.  In  A  and 
B  the  cylinder  was  unjacketed;  in  C  and  D  it  was  jacketed.  The  beneficial  in- 


FIG.   199.     Art.  458.  — Initial   Pressure    and 
Back  Pressure. 


FIG.  200.    Art.  458.  —  Effects  of  Jacket- 
ing and  Superheating. 


fluence  of  the  jackets  is  clearly  shown,  but  not  the  expenditure  of  heat  in  the 
jacket.  The  steam  consumption  in  the  four  cases  was  45.6,  28.4,  27.25  and 
20.9  Ib.  per  Ihp-hr.,  respectively. 


Without 


\ 


MULTIPLE  EXPANSION 

459.  Desirability  of  Complete  Expansion.  It  is  proposed  to  show  that  a  large 
ratio  of  expansion  is  from  every  standpoint  desirable,  excepting  as  it  is  offset  by 
increased  cylinder  condensation  ;  and  to  suggest  multiple  expansion  as  a  method 
for  attaining  high  efficiency  by  making  such  large  ratio  practically  possible. 

From  Art.  446,  it  is  obvious  that  the  maximum  work  obtainable  from  a  cylinder  is 
a  function  solely  of  the  initial  pressure,  the  back  pressure,  and  the  ratio  of  expan- 
sion. In  a  non-conducting  cylinder,  maximum  efficiency  would  be  realized  when 
the  ratio  of  expansion  became  a  maximum  between  the  pressure  limits. 
expansion,  increase  of  initial  pressure  very  slightly,  if 
at  all,  increases  the  efficiency.  Thus,  in  Fig.  201, 
the  cyclic  work  areas  abed,  aefg,  ahij,  would  all  be 
equal  if  the  line  X  Y  followed  the  law  pc  =  PV. 
As  the  actual  law  (locus  of  points  representing 
steam  dry  at  cut-off)  is  approximately, 

,»«  =  pj-H, 

the  work  areas  increase  slightly  as  the  pressure  in- 
creases; but  the  necessary  heat  absorption  also 
increases,  and  there  is  little  or  no  net  gain.  The 
thermodynamic  advantage  of  high  initial  pressure  is 
realized  only  when  the  ratio  of  expansion  is  large. 

By  condensing  the  steam  as  it  flows  from  the  engine,  its  pressure  may  be  re- 
duced from  that  of  the  atmosphere  to  an  absolute  pressure  possibly  13'  Ib.  lower. 
The  cyclic  work  area  is  thus  increased  :  and  since  the  reduction  of  pressure  is  ac- 
companied by  a  reduction  in  temperature,  the  potential  efficiency  is  increased. 
Figure  202  shows,  however,  that  the  percentage  gain  in  efficiency  is  small  with  no 


FIG.  201.    Art.  459.  — Non- 
expansive  Cycles. 


276 


APPLIED  THERMODYNAMICS 


expansion,  increasing  as  the  expansion  ratio  increases.     Wide  ratios  of  expansion  are 

from  all  of  these  standpoints  essential  to  efficiency. 

We  have  found,  however,  that  wide  ratios  of 
expansion  are  associated  with  such  excessive  losses 
from  condensation  that  a  compromise  is  necessary, 
and  that  in  practice  the  best  efficiency  is  secured 
with  a  rather  limited  ratio.  The  practical  attain- 
ment of  large  expansive  ratios  without  correspond- 
ing losses  by  condensation  is  possible  by  multiple 
expansion.  By  allowing  the  steam  to  pass  suc- 
cessively through  two  or  more  cylinders,  a  total 
expansion  of  10  to  25  may  be  secured,  with  condensa- 
tion losses  such  as  are  due  to  much  lower  ratios. 


FIG.  202.    Art.  459.  — Gain  due 
to  Vacuum. 


460.  Condensation  Losses  in  Compound  Cylinders.     The  range  of  pres- 
sures, and  consequently  of  temperatures,  in  any  one  cylinder,  is  reduced 
by  compounding.     It  may  appear  that  the  sum  of  the  losses  in  the  two 
cylinders  would  be  equal  to  the  loss  in  a  single  simple  cylinder.     Three 
considerations  may  serve  to  show  why  this  is  not  the  case : 

(a)  Steam  reevaporated  during  the  exhaust  stroke  is  rendered  avail- 
able  for   doing  work  in   the   succeeding  cylinder,  whereas  in  a  simple 
engine  it  merely  causes  a  resistance  to  the  piston. 

(b)  Initial  condensation  is  decreased  because  of  the  decreased  fluctua- 
tion in  wall  temperature. 

(c)  The  "range  of  temperature  in  each  cylinder  is  half  what  it  is  in  the 
simple  cylinder,  but  the  whole  wall  surface  is  not  doubled. 

461.  Classification.     Engines  are  called  simple,  compound,  triple,  or  quadruple, 
according  to  the  number  of  successive  expansion  stages,  ranging  from  one  to  four. 
A  multiple-expansion  engine  may  have  any  number  of  cylinders;  a  triple  expan- 
sion engine  may,  for  example,  have  five  cylinders,  a  single  high -pressure  cylinder 
discharging  its  steam  to  two  succeeding  cylinders,  and  these  to  two  more.     In  a 
multiple-expansion  engine,  the  first  is  called  high-pressure  cylinder  and  the  last 
the  low-pressure  cylinder.      The  second  cylinder  in  a  triple  engine  is  called  the 
intermediate;    in  a  quadruple  engine,  the  second  and  third  are  called  the  first 
intermediate  and  the  second  intermediate  cylinders,  respectively.     Compound  en- 
gines having  the  two  cylinders  placed  end  to  end  are  described  as  tandem ;  those 
having  the  cylinders  side  by  side  are  cross-compound.     This  last  is  the  type  most 
commonly  used  in  high-grade  stationary  plants  in  this  country.     The  engines  may 
be  either  horizontal  or  vertical ;  the  latter  is  the  form  generally  used  for  triples  or 
quadruples,  and  in  marine  service.     Sometimes  some  of  the  cylinders  are  horizon- 
tal and  others  vertical,  giving  what,  in  the  two-expansion  type,  has  been  called  the 
angle  compound.     Compounding  may  be  effected  (as  usually)  by  using  cylinders  of 
various  diameters  and  equal  strokes;  or  of  equal  diameters  and  varying  strokes, 
or  of   like  dimensions   but   unequal  speeds  (the  cylinders  driving  independent 
shafts),  or  by  a  combination  of  these  methods. 


WOOLF  COMPOUND  ENGINE 


277 


462.  Incidental  Advantages.  Aside  from  the  decreased  loss  through  cylinder 
condensation,  multiple-expansion  engines  have  the  following  points  of  superiority  : 

(1)  The  steam  consumed  in  filling  clearance  spaces  is  less,  because  the  high- 
pressure  cylinder  is  smaller  than  the  cylinder  of  the  equivalent  simple  engine. 

(2)  Compression  in  the  high-pressure  cylinder  may  be  carried  to  as  high  a 
pressure  as  is  desirable  without  beginning  it  so  early  as  to  greatly  reduce  the  work 
area. 

(3)  The  low-pressure  cylinder  need  be  built  to  withstand  a  fraction  only  of 
the  boiler  pressure ;   the  other  cylinders,  which  carry  higher  pressures,  are  com- 
paratively small. 

(4)  In  most  common  types,  the  use  of  two  or  more  cylinders  permits  of  using 
a  greater  number  of  less  powerful  impulses  on  the  piston  than  is  possible  with  a 
single  cylinder,  thus  making  the  rotative  speed  more  uniform. 

(5)  For  the  same  reason,  the  mechanical  strains  on  the  crank  pin,  shaft,  etc., 
are  lessened  by  compounding. 

These  advantages,  with  that  of  superior  economy  of  steam,  have  led  to  the 
general  use  of  multiple  expansion  in  spite  of  the  higher  initial  cost  which  it  en- 
tails, wherever  steam  pressures  exceed  100  Ib. 

463.  Woolf  Engine.  This  was  a  form  of  compound  engine  originated  by  Horn- 
blower,  an  unsuccessful  competitor  of  Watt,  and  revived  by  Woolf  in  1800,  after 
the  expiration  of  Watt's  principal  patent. 
Steam  passed  directly  from  the  high  to  the 
low-pressure  cylinder,  entering  the  latter 
while  being  exhausted  from  the  former. 
This  necessitated  having  the  pistons  either 
in  phase  or  a  half  revolution  apart,  and 
there  was  no  improvement  over  any  other 
double-acting  engine  with  regard  to  uni- 
formity of  impulse  on  the  piston.  Figure 
203  represents  the  ideal  indicator  diagrams.  FlG  203.  Arts. 463, 466.—  Woolf  Engine. 
A  BCD  is  the  action  in  the  high-pressure 

cylinder,  the  fall  of  pressure  along  CD  being  due  to  the  increase  in  volume  of 
the  steam,  now  passing  into  the  low-pressure  cylinder  and  forcing  its  piston  out- 
ward. EFGH  shows  the  action  in  the  low-pres- 
sure cylinder;  steam  is  entering  continuously 
throughout  the  stroke  along  EF.  By  laying  off 
MP  =  LK,  etc.,  we  obtain  the  diagram  TABRS, 
representing  the  changes  undergone  by  the  steam 
during  its  entire  action.  This  last  area  is  ob- 
viously equal  to  the  sum  of  the  areas  A  BCD 
and  EFGH.  Figure  204,  from  Ewing  (19) 
shows  a  pair  of  actual  diagrams  from  a  Woolf 
engine,  the  length  of  the  diagrams  representing 
the  stroke  of  the  pistons  and  not  actual  steam  volumes.  The  low-pressure  dia- 
gram has  been  reversed  for  convenience.  Some  expansion  in  the  low-pressure 


FIG.  204.    Art.  463,  Prob.  31.  —Dia- 
grams from  Woolf  Engine. 


278  APPLIED  THERMODYNAMICS 

cylinder  occurs  after  the  closing  of  the  high-pressure  exhaust  valve  at  «.  Some 
loss  of  pressure  by  wiredrawing  in  the  passages  between  the  two  cylinders  is 
clearly  indicated. 

464.  Receiver  Engine.     In  this  more  modern  form  the  steam  passes 
from  the  high-pressure  cylinder  to  a  closed  chamber  called  the  receiver, 
and  thence  to  the  low-pressure  cylinder.     The  receiver  is  usually  an  inde- 
pendent vessel  connected  by  pipes  with  the  cylinders ;  in  some  cases,  the 
intervening  steam  pipe  alone  is  of  sufficient  capacity  to  constitute  a  re- 
ceiver.    Receiver  engines  may  have  the  pistons  coincident  in  phase,  as  in 
tandem  engines,  or  opposite,  as  in  opposed  beam  engines,  or  the  cranks  may 
be  at  an  angle  of  90°,  as  in  the  ordinary  cross-compound.     In  all  cases  the 
receiver  engine  has  the  characteristic  advantage  over  the  Woolf  type  that 
the  low-pressure  cylinder  need  not  receive  steam  during  the  whole  of  the 
working  stroke,  but  may  have  a  definite  point  of  cut-off,  and  work  in  an 
expansive  cycle.     The  distribution  of  work  between  the  two  cylinders,  as 
will  be  shown,  may  be  adjusted  by  varying  the  point  of  cut-off  on  the  low- 
pressure  cylinder  (Art.  467). 

465.  Drop.     The  fall  of  pressure  occurring  at  the  end  of  expansion 
is  termed  drop.     Its  thermodynamic  disadvantage  and  practical  necessity 
have  been  discussed  (Arts.  418,  447).     In  a  compound  engine,  drop  in 
the  high-pressure  cylinder  has  the  additional  effect  of  seriously  influenc- 
ing the  amount  of  work  done.     With  no  such  drop  the  combined  ideal 
diagrams  of  a  receiver  engine  would  be  precisely  the  same  as  that  of  a 
simple  cylinder  with  the  same  amount  of  expansion. 

466.  Combined  Diagrams.    Figure  205  shows  the  ideal  diagrams  from  a  tandem 
receiver  engine.     Along  CD,  as  along  CD  in  Fig.  203,  expansion  into  the  low- 
pressure  cylinder  is  taking  place.     The  cor- 
responding line  on  the  low-pressure  diagram 
is  FG.     At  G  the  supply  of  steam  is  cut  off 
from  the  low-pressure  cylinder,  after  which 
hyperbolic     expansion     occurs    along    Gff. 
Meanwhile,  the  exhaust  from  the  high-pres- 
sure cylinder  is  discharged  to  the  receiver; 
and  since  a  constant  quantity  of  steam  must 
now  be   contained  in   the  decreasing  space 

FIG.  205.     Art.  466.  — Combined  Dia-     between   the   piston    and   the    cylinder  and 
grams,  Tandem  Receiver  Engine.  receiver  walls,  some  compression  occurs,  giv- 

ing the  line  DE.  The  pressure  of  the  re- 
ceiver steam  remains  equal  to  that  at  E  after  the  high-pressure  exhaust  valve 
closes  (at  E)  and  while  the  high-pressure  cylinder  continues  the  cycle  along 
EABC.  If  the  pressure  at  C  exceeds  that  at  E,  then  there  will  be  some  drop. 
As  drawn,  the  diagram  shows  none.  If  cut-off  in  the  low-pressure  cylinder 


TANDEM   RECEIVER   ENGINE  279 

occurred  later  in  the  stroke,  the  line  DE  would  be  lowered,  Pc  would  exceed  PE, 
and  drop  would  be  shown. 

An  important  advantage  of  the  receiver  engine  is  here  evident.  The  intro- 
duction of  cut-off  in  the  low-pressure  cylinder  raises  the  lower  limit  of  tempera- 
ture in  the  high-pressure  cylinder  from  D  in  Fig.  203  to  D  in  Fig.  205.  This 
reduced  range  of  temperature  decreases  cylinder  condensation. 

467.  Adjustment  of  Work.  Figure  206  shows  the  diagrams  as 
they  appear  with  drop.  Now  if  cut-off  in  the  low-pressure  cylinder 
be  made  ,to  occur  a  little  earlier  in  the  stroke,  the  pressures  at  D 
and  along  the  compression  path  DE 
would  be  increased,  and  the  work  area 
of  the  high-pressure  cycle  would  be  de- 
creased. Tbe  initial  pressure  in  the  low- 
pressure  cylinder  (which  depends  upon 
PB  as  well  as  Pc)  would  be  increased. 
The  tendency  toward  a  reduction  of  area 


of  the  low-pressure  cycle  by  earlier  cut-  FIG.  206.    Art.  467.  —  Receiver  En- 

^    •  .7  rr    j.    T        J.-L  7  gine  with  Drop. 

off  is  more  than  offset   by    the  increased 

initial  pressure.  The  fact  is  that  the  total  work  of  the  engine  is 
scarcely  affected  by  a  change  in  low-pressure  cut-off.  The  low-pres- 
sure work  area  increases  to  almost  precisely  the  same  extent  that 
the  high-pressure  area  decreases.  We  have  thus  the  peculiar  re- 
sult that  with  earlier  cut-off  the  low-pressure  cylinder  performs  a 
greater  proportion  of  the  total  work.  Earlier  cut-off  decreases  drop. 
The  problem  of  compound  engine  design  is  to  adjust  the  cylinder 
and  receiver  volumes  and  the  point  of  low-pressure  cut-off  so  that 
the  desired  amount  of  drop  may  be  secured  along  with  practically 
equal  distribution  of  work  between  the  two  cylinders. 

468.  Assumptions.     In  some  cases,  the  cylinders  are  so  proportioned  as  to 
make  the  range  of  temperatures  the  same  in  each.     This  usually  involves  the 
performance  of  very  nearly  equal  amounts  of  work ;  the  equalization  of  work 
areas  is  the  more  usual  aim.     The  question  of  the  desirable  amount  of  drop  will 
be  considered  later.     For  the  present,  we  will  assume  it  to  be  zero.     In  some 
marine  engines,  with  valve  gears  which  involve  a  rather  late  low-pressure  cut-off 
at  running  speeds,  the  desired  flexibility  cannot  be  obtained  without  a  consider- 
able amount  of  drop  between  the  cylinders. 

469.  Application  to  Tandem  Compound.     In  Fig.  207,  let  A  BCD  be  a  portion 
of   the  indicator   diagram  of   the    high-pressure  cylinder  of  a  tandem   receiver 


280  APPLIED  THERMODYNAMICS 

engine,  release  occurring  at  C.  At  this  point,  the  whole  volume  of  steam  consists 
of  that  in  the  receiver  plus  that  in  the  high-pressure  cylinder.  Let  the  receiver 
volume  be  represented  by  the  distance  CX.  Then  the  hyperbolic  curve  XY  may 

represent  the  expansion  of  the 
steam  between  the  states  C  and 
D,  and  by  deducting  the  constant 
volumes  CX,  LR,  MZ,  etc.,  we  ob- 
tain the  curve  CG,  representing 
the  expansion  of  the  steam  in  the 
two  cylinders.  For  no  drop,  the 
pressure  at  the  end  of  compression 
into  the  receiver  must  be  equal 
to  that  at  C.  We  thus  find  the 
point  E,  and  draw  EF,  the  ad- 
mission line  of  the  low-pressure 
cylinder,  such  that  ac  +  ad  =  ae, 
etc.;  the  abscissa  of  cC  being  to 

that  of  Ed  in  the  same  ratio  as 
FIG.  207.     Art.  469. -Elimination  of  Drop,  Tandem    ^    Q£    the  ctive        Under 

Receiver  Engine.  n         .   ...         ^  _. 

volumes.      By    plotting    ED   we 

find  the  point  D  at  its  intersection  with  CD.  A  horizontal  projection  from  D 
to  EF  gives  F.  The  point  F  is  then  the  required  point  of  cut-off  in  the  low- 
pressure  cylinder.  The  diagram  EFSHI  may  be  completed,  the  curve  FS  being 
hyperbolic. 

470.  Cranks  at  Right  Angles.  In  Fig.  208,  let  abC  be  a  portion  of  the  high- 
pressure  diagram,  release  occurring  at  C.  Communication  is  now  opened  with  the 
receiver.  Let  the  receiver  volume  be  laid  off  as  Cd,  and  let  de  be  a  hyperbolic 
curve.  Then  the  curve  Cf,  the  volume  of  which  at  any  pressure  is  Cd  less  than 
that  of  de,  represents  the  path  in  the  high-pressure  cylinder.  This  continues  until 
admission  to  the  low-pressure  cylinder  occurs  at  g.  The  whole  volume  of  steam  is 
now  made  up  of  that  in  the  two  cylinders  and  the  receiver;  the  volumes  in  the 
cylinders  alone  are  measurable  out  to/C.  In  Fig.  209,  lay  off  hi  =  1C  and/ft  such 
that  jk  ~  hi  is  equal  to  the  ratio  of  volumes  of  low-  and  high-pressure  cylinder. 
At  the  point  C  of  the  cycle,  the  high -pressure  crank  is  at  i,  the  low-pressure  crank 
90°  ahead  or  behind  it.  When  the  high-pressure  crank  has  moved  from  i  to  rn, 
the  volume  of  steam  in  that  cylinder  is  represented  by  the  distance  hn,  the  low- 
pressure  crank  is  at  o  and  the  volume  of  steam  in  the  low-pressure  cylinder  is 
represented  by  pk.  Lay  off  qr,  in  Fig.  208,  distant  from  the  zero  volume  line  al 
by  an  amount  equal  to  Jin  +  pk.  Draw  the  horizontal  line  ts.  Lay  off  tu  =  hn  and 
tv  =  us  =pk.  Then  u  is  a  point  on  the  high-pressure  exhaust  line  and  v  is  a  point 
on  the  low-pressure  admission  line.  Similarly,  we  find  corresponding  crank  posi- 
tions w  and  x,  and  steam  volumes  hy  and  zk,  and  lay  off  AB  =  hy  +  zk,  Ac-  hy, 
A  D  =  cB  =  zk,  determining  the  points  c  and  D.  The  high-pressure  exhaust  line 
guc  is  continued  to  some  distance  below  I.  For  no  drop,  this  line  must  terminate 
at  some  point  such  that  compression  of  steam  in  the  high-pressure  cylinder  and 
receiver  will  make  I  the  final  state.  At  I  the  high-pressure  cylinder  steam  volume 


CROSS-COMPOUND  ENGINE 


281 


is  zero ;  all  the  steam  is  in  the  receiver.  Let  IE  represent  the  receiver  volume 
and  EF  a  hyperbolic  curve.  Draw  IG  so  that  at  any  pressure  its  volumes  are 
equal  to  those  along  EF,  minus  the  constant  volume  IE.  Then  H,  where  IG 
intersects  yuc,  is  the  state  of  the  high-pressure  cycle  at  which  cut-off  occurs  in 
the  low-pressure  cylinder.  By  drawing  a  horizontal  line  through  H  to  intersect 
vD,  we  find  the  point  of  cut-off  /  on  the  low-pressure  diagram.  If  we  regard  the 
initial  state  as  that  when  admission  occurs  to  the  low-pressure  cylinder,  then  at 

low-pressure  cut-off  the  high-pressure  cylinder  will  have  completed  the  pro- 

t  O 

portion  of  a  full  stroke.  Modifications  of  this  construction  permit  of  determining 
the  point  of  cut-off  for  no  drop  in  triple  or  quadruple  engines  with  any  phase 
relation  of  the  cranks. 

471.  Analytical  Method  :  Tandem  Engine.  Let  the  volume  of  the  high-pressure 
cylinder  be  taken  as  unity,  that  of  the  receiver  as  R,  that  of  the  low-pressure  cylinder 
as  L.  Let  x  be  the  fraction  of  its  stroke  completed  by  the  low-pressure  piston  at 
cutroff,  and  let  p  be  the  pressure  at  release  from  the  high-pressure  cylinder,  equal 
to  the  receiver  pressure  at  the  moment  of  admission  to  the  low-pressure  cylinder. 
The  volume  of  steam  at  this  moment  is  1  +  R :  at  low-pressure  cut-off,  it  is 
1  +  R  +  xL  —  x.  If  expansion  follows  the  law  pv  =  P  V,  and  P  be  the  pressure  in 
the  low-pressure  cylinder  at  cut-off, 

1  4-  R 
p(\  +  R)  =P(1+  R  +  xL  -a:),  or  P=p 


1  +  R  +  xL  -  x 

The  remaining  quantity  of  steam  in  the  high-pressure  cylinder  and  receiver  has 
the  volume  1  —  x  +  /?,  which,  at  the  end  of  the  stroke,  will  have  been  reduced  to 
R.  If  the  pressure  at  the  end  of  the  stroke  is  to  be  p,  then 


Combining  the  two  values  of  P,  we  find 


x  — 


R  4-1 
RL  +  l 


472.    Cross-compound  Engine.     In  this  case,  the  fraction  of  the  stroke  completed 
at  low-pressure  cut-off  is  different  for  the  two  cylinders.     Let  X  be  the  proportion 


FIG.  208.     Arts.  470,  472,  473.  —Elimination  of  Drop,  Cross-compound  Engine. 


282 


APPLIED  THERMODYNAMICS 


of  the  high-pressure  stroke  occurring  between  admission  and  cut-off  in  the  low- 
pressure  cylinder.  Proceeding  as  before,  the  volume  of  the  steam  at  low-pressure 
admission  is  0.5  +  R,  and  that  at  low-pressure  cut-off  is  0.5  -  X  +  R  +  xL.  The 
volume  of  steam  in  the  high-pressure  cylinder  and  the  receiver  at  the  end  of  the 
high-pressure  exhaust  stroke  is  R  ;  the  volume  just  after  low-pressure  cut-off  occurs 
is  0.5  —  X  4-  R.  The  volume  at  the  beginning  of  exhaust  from  the  high-pressure 
cylinder  is  1  4-  R.  In  Fig.  208,  let  the  pressure  at  C  and  I  be  p ;  let  that  at  g  be  P. 
Then  ,„  .  ^x  n,^  „  .  „,  _  n  1  4-  R 


Let  the  pressure  at  H  be  Q  :  then 


P(0.5  + 


or 


^0.5 +  * 

=  Q(Q.5-X  +  R  +  xL), 

!)  P(l  4-  R} 


(0.5-X  +  R  +  *Z)(0.5  +  Jft)      O.o-X  +  R+xL 

-X  +  R)  =    (14- JQ(0.5-Ar  +  fl)  . 
P  R(0.5  -X  4-  R  +  xL)  ' 

whence, 

In  Fig.  209,  we  have  the  crank 
circles  corresponding  to  the 
discussed  movements.  If  Ow 
and  Ox  are  at  right  angles, 
then  for  a  high-pressure  pis- 
ton displacement  Oy,  we  have 
the  corresponding  low-pres- 
sure displacement^.  If  these 
displacements  be  taken  as  at 
low-pressure  cut-off,  then 

~  Yi  a          ~Jk' 
We  may  also  draw  OwP,  PQ> 
and  write  X  =  ^-     In  the 


FIG.  209.     Arts.  470,  472.  — Crank  Circles  and  Piston 
Displacements. 


triangles    OPQ,     Oxz,    OQ  = 


xz=jk .  X,  xz*+  0?=  Ox*,  and 
(jk-  X)2+  fj~  -  x-jk\*=  (7-]  '  whence  X  -  Vx-x'2.  Substituting  this  value  in 
Equation  (A),  we  find  R  (xL  —  1)  =  0.5  —  Vx  —  x'2  as  the  condition  of  no  drop. 

473.  Practical  Modifications.  The  combined  diagrams  obtained  from  actual 
engines  conform  only  approximately  to  those  of  Figs.  207  and  208.  The  receiver 
spaces  are  usually  so  large,  in  proportion  to  the  volume  of  the  high-pressure 
cylinder,  that  the  fluctuations  of  pressure  along  the  release  lines  are  scarcely  notice- 
able. The  fall  of  pressure  during  admission  to  the  low-pressure  cylinder  is,  how- 
ever, nearly  always  evident.  Marked  irregularities  arise  from  the  angularity  of  the 


COMBINED   DIAGRAMS 


283 


connecting  rod.  In  assuming  crank  positions  and  piston  displacements  to  corre- 
spond, we  have  tacitly  assumed  the  rod  to  be  of  infinite  length;  in  practice,  it  seldom 
exceeds  five  or  six  times  the  length  of  the  crank.  The  receiver  volume  is  made 
from  1  to  I1  times  that  of  the  cylinder  by  which  it  is  supplied.  Its  size  has  theo- 
retically no  effect  on  the  efficiency  of  the  engine.  We  have  assumed  all  expansion 
paths  to  be  hyperbolic ;  an  assumption  not  strictly  justified  for  the  conditions  con- 
sidered ;  and  we  have  ignored  modifying  influences  due  to  clearance.  Some  designers, 
particularly  in  the  case  of  marine  engines,  aim  at  equalizing  the  maximum  pressures 
on  cranks  rather  than  at  equalization  of  load;  careful  allowance  must  then  be  made 
for  clearance  and  compression. 

474.  Losses  in  Multiple-expansion  Engines.  Aside  from  those  already  discussed 
in  connection  with  simple  engines,  the  losses  in  a  multiple-expansion  engine 
include  that  due  to  pressure  drop,  if  any,  between  the  high-pressure  cylinder 
and  the  receiver,  and  that  due  to  friction  in  the  intermediate  passages.  These 
are  partially  offset  by  superheating  resulting  from  the  wiredrawing. 

475.  Combination  of  Actual  Diagrams  :  Diagram  Factor.  Figure  210  shows  the 
high-  and  low-pressure  diagrams  from  a  small  compound  engine.  These  are  again 


L.P. 


FIG.    210.      Art.  475.  —  Compound    Engine 
Diagrams. 


FIG.    211.      Art.  475.  —  Compound    Engine 
Diagrams  Combined. 


shown  in  Fig.  211,  in  which  the  lengths  of  the  diagrams  are  proportioned  as  are 
the  cylinder  volumes,  the  pressure  scales  are  made  equal,  and  the  proper  amounts 
of  setting  off  for  clearance  (distances  a  and  6)  are  regarded.  The  cylinder  feed 
per  single  stroke  was  0.0498  lb.,  the  cushion  steam  in  the  high-pressure  cylinder 
0.0074  lb.,  and  that  in  the  other  cylinder  0.0022  lb.  No  single  saturation  curve 
is  possible  ;  the  line  ss  is  drawn  for  0.0572  lb.  of  steam,  and  SS  for  0.0520  lb.  As 
in  Art.  453,  we  may  obtain  equivalent  diagrams  with  the  cushion  steam  eliminated. 
In  Fig.  212.  the  single  curve  SS  then  represents  saturation  for  0.0498  lb.  of  steam. 
The  areas  of  the  diagrams  are  unaltered,  and  correctly  measure  the  work  done ; 
they  may  be  transferred  to  the  entropy  plane  as  in  Art.  454.  The  moisture  present 
at  any  point  during  expansion  is  still  represented  by  the  distance  cd,  correspond- 
ing to  the  distance  similarly  marked  in  Fig.  211.  In  Fig.  213  this  construction 
has  been  applied  to  a  triple-expansion  engine,  the  first  diagram  showing  the 
action  when  unjacketed,  and  the  second,  when  jackets  are  used.  The  drying 
influence  of  the  jackets  is  clearly  shown.  The  ratio  of  the  area  of  the  combined 


284 


APPLIED  THERMODYNAMICS 


actual  diagrams  to  that  of  the  Rankine  cycle  through  the  same  extreme  limits 
of  pressure  and  with  the  same  ratio  of  expansion  is  the  diagram  factor,  the  value 


UNJACKETED 


FIG.  212.    Art.  47n.—  Combined  Diagrams 
for  Cylinder  Feed. 


FIG.  213.    Art.  475.  — Triple  Engine 
Diagrams. 


of  which  may  range  from  0.70  upward,  being  higher  than  in  simple  engines  having 
the  same  total  ratio  of  expansion,  but  not  higher  than  in  the  simple  engines  of 
ordinary  practice  (Art.  459). 

476.  Compound  Engine  Capacity.  If  e  be  the  real  ratio  of  expansion  in  the 
high-pressure  cylinder,  and  L  the  ratio  of  cylinder  volumes,  the  total  real  ratio  of 
expansion  is  E  =  eL.  If  i  is  the  clearance  in  the  high-pressure  cylinder  expressed 
as  a  fraction  of  the  volume  of  that  cylinder,  and  k  is  the  apparent  ratio  of  expan- 


sion therein,  we  may  show  that  k  =  —    — ,  E  = 


kL 


-i  +  ik 


The  total  real  ratio  of 


expansion  is  thus  independent  of  the  point  of  cut-off  on  the  low-pressure  cylinder.  It 
ranges  usually  from  10  to  25,  increasing  as  the  number  of  expansive  cylinders 
increases.  In  compound  engines  it  is  most  commonly  16. 

Given  the  same  initial  pressure  and  back  pressure,  total  real  ratio  of  expansion, 
and  diagram  factor,  the  low-pressure  cylinder  volume  of  a  multiple-expansion  engine 
is  obviously  the  same  as  that  of  the  simple  engine  cylinder  of  equal  capacity.  It  is 
common  practice  to  establish  mean  receiver  pressures  which  will  at  normal  load, 
without  drop,  give  equal  distribution  of  work  between  the  cylinders.  If  the  vari- 
ous computed  mean  effective  pressures  are  then  divided  by  the  ratio  of  low-pressure 
cylinder  volume  to  that  of  the  cylinder  under  consideration,  and  the  quotients 
added,  we  have  the  "mean  effective  pressure  referred  to  the  low-pressure  cylinder." 
The  capacity  may  be  calculated  from  this  and  from  the  dimensions  and  piston 
speed  of  that  cylinder. 

The  size  of  the  low-pressure  cylinder  being  determined  as  S,  that  of  tha  high- 
pressure  cylinder  is  — S,  the  minimum  value  of  which  is  —  The  value  of  E 
E  .  E 

may  be  adjusted  at  will  by  varying  the  point  of  high-pressure  cut-off,  regardless  of 
the  cylinder  ratio.  From  this  standpoint,  then,  the  size  of  the  high-pressure 
cylinder  is  without  influence  on  the  efficiency.  Non-condensing  compound  engines 
usually  have  a  low-pressure  cylinder  from  3  to  4  times  larger  than  the  high-pres- 
sure cylinder.  With  condensing  engines,  the  ratio  is  usually  4  to  6,  increasing 
with  the  boiler  pressure.  In  triple  engines,  the  ratios  are  from  1 :2.0  :  2.0  up  to 


DESIGN   OF   COMPOUND   ENGINE 


285 


1 : 2.5  :  2.5  in  stationary  practice.  In  quadruple  engines  the  ratios  are  successively 
from  2.0  to  2.5  : 1.  The  use  of  two-stage  or  compound  expansion  is  common  prac- 
tice everywhere.  Triple  and  quadruple  engines,  in  which  much  higher  initial 
pressures  are  desirable,  are  used  mostly  in  marine  service.  In  stationary  applica- 
tions, a  few  of  these  high-stage  engines  are  in  use,  with  excellent  results  as  to  fuel 
economy ;  but  it  is  only  where  the  cost  of  fuel  or  the  load  factor  is  high  or  capital 
charges  low  that  they  have  to  any  considerable  extent  been  found  more  profitable, 
commercially,* than  the  compound  engine. 

477.  Specimen  Design.  Let  the  engine  develop  1000  Ihp.  at  100  r.  p.  m., 
ivith  pressure  limits  of  150  and  2  Ib.  absolute  and  a  ratio  of  expansion  of 16, 
the  piston  speed  being  800  ft.  per  minute. 

In  a  simple  engine,  the  m.  e.  p.  would  be  (Art.  446)  -          t"  °gg — *  —  2 

800 
=  88.5,  and  the  stroke  -      —  =  4  feet  or  48  inches.     We  will  ignore  the 

2  x  100 

diagram  factor  in  order  to  more  rigorously  compare  sizes;  the  area  of  the 
cylinder  of  the  simple  engine  is  then  (33,000  x  1000)  -r-  (33.5  x  800)=  1280 
square  inches. 

In  the  compound  engine,  let  the  cylinder  ratio  be  first  established,  say 
as  4.  The  mean  effective  pressure  of  the  combined  diagrams  is  33.5.  If 
we  assign  half  of  this  to  the  low-pressure  cylinder,  its  area  must  be  (500  x 

'°  x  800    =  1280  square   inches,  precisely   that  of  the  simple 

cylinder.     The  m.  e.p.  in  the  high-pressure  cylinder  referred  to  the  loic-pres- 

335  33  5 

sure  cylinder  (Art.  476)  is  also  -  — :  its  actual  m.  e.p.  is  then  '-^-  x4  =  67, 


33,000)  -r- 


|§M 


and  its  area  is  (33,000  x  500)  -=-  (67  x  800)  =  307%  square  inches :  or,  more 
briefly,  -    —  =  307^.     This  gives  an  engine  in  which  the  work  distribu- 


tion  with  no  drop  may  be  unequal.*  If  actual  diagram  factors  are  intro- 
duced, the  low-pressure  cylinder  of  the  compound  will  differ  somewhat  in 
size  from  the  cylinder  of  the  equivalent 
simple  engine. 

478.  Governing  Compound  Engines.  It 
has  been  shown  (Art.  467)  that  earlier  cut- 
off on  the  low-pressure  cylinder  relieves  the 
high-pressure  cylinder  of  some  of  its  propor- 
tion of  the  load.  Figure  214  shows  further 
that  delayed  cut-off  on  the  high-pressure  cyl- 
inder greatly  increases  the  work  done  in  the 
low-pressure  cylinder,  while  only  slightly 


FIG.  214. 


Art.   478.  — Effect  of  Low- 
pressure  Cut-off. 


*  High  cylinder  ratios,  with  equal  work  distribution,  are  possible  only  when  the 
total  number  of  expansions  is  high.  It  is,  of  course,  permissible  to  design  the  engine 
so  that  each  cylinder  does  half  the  work.  See  problem  27,  page  314. 


286  APPLIED  THERMODYNAMICS 

increasing  its  own  work  area.  When  the  load  increases  in  an  engine  which  is  gov- 
erned by  adjustment  of  the  high-pressure  cut-off  only,  equality  of  work  distribu- 
tion becomes  impossible.  For  economy,  the  governor  should  control  the  cut-off 
on  both  cylinders,  making  it  later  on  both  as  the  load  increases,  but  not  in  the 
same  proportion. 

Variation  of  cut-off  in  the  low-pressure  cylinder  permits  of  adjustment  of  the 
division  of  work  between  the  cylinders,  irrespective  of  the  sizes ;  but  absence  of 
drop  is  simultaneously  possible  only  when  the  cylinder  ratios  are  correct.  Adjust- 
ment of  low  pressure  cut-off  to  eliminate  drop,  in  badly  proportioned  cylinders, 
results  in  an  unequal  distribution  of  work. 

To  summarize :  the  power  of  the  engine  is  varied  by  varying  the 
high-pressure  cut-off ;  wide  ranges  of  capacity  are  obtainable  only 
when  the  high-pressure  cylinder  is  comparatively  large :  the  distri- 
bution of  the  work  is  kept  uniform  by  varying  the  low-pressure  cut- 
off ;  and  this  results  in  a  loss  of  efficiency  due  to  the  excessive  drop 
unless  the  cylinder  proportions  are  right. 

479.  The  Drop  Controversy.     Thus  far,  we  have  treated  the  subject  from  the 
standpoint  that  maximum  efficiency  is  attained  with  a  zero  drop  in  pressure  at 
high-pressure  release.     This  is  the  orthodox  view,  maintained  in  this  country  by 
many  engineers,  and  almost  universally  followed  in  European  practice.     Some 
authorities  have  contended  that  a  limited  amount  of  drop  is  both  practically  and 
thermodynamically  desirable  (21).     From  Art.  447,  it  is  obvious  that  in  a  single 
cylinder,  expansive  ratios  exceeding  certain  limits  become  undesirable  on  account 
of  excessive  cylinder  condensation :  in  such  cylinders,  a  constant  volume  drop  at 
the  end  of  the  stroke  is  always  permitted.     In  a  compound  engine,  drop  decreases 
the  diagram  factor  of  the  combined  diagram  :  and  it  has  been  usually  regarded  as 
objectionable  on  any  but  the  last  cylinder  of  the  series.     The  aim  of  designers  has 
been  to  make  the  actual  expansion  line  coincide  with  the  hyperbolic  curve  as 
closely  as  possible ;  and  for  this  reason  the  harmful  influence  of  drop  has  possibly 
been  overemphasized,  and  the  argument  in  its  favor  disregarded.     There  is,  in 
fact,  a  special  argument  for  drop  in  multiple-expansion  engines,  from  the  fact 
that  unresisted  expansion  leads  to  a  drying  of  the  steam,  which  exerts  a  beneficial 
effect  in  the  succeeding  cylinder. 

480.  Intermediate   Compounds.     Tests  by  Kockwood  (22)  of  a  triple 
engine  in  which  the  intermediate  cylinder  was  cut  out,  permitting  of  run- 
ning the  high-  and  low-pressure  cylinders  as  a  compound  with  the  high 
cylinder  ratio  of  5.7  to  1,  give  the  surprising  result  that  with  the  same 
initial  pressure  and  expansive  ratio,  the  compound  was  more  economical 
than  the  triple.     This  was  a  small  engine,  with  large  drop.     The  pointing 
out  of  the  fact  that  the  conditions  were  unduly  favorable  to  the  compound 
as  compared  with  the  triple  did  not  explain  the  excellent  economy  of  the 
former  as  compared  with  all  engines  of  its  class.     Somewhat  later,  excep- 


REHEATERS 


287 


tionally  good  results  were  obtained  by  Barrus  (23)  with  a  compound 
engine  having  the  extraordinary  cylinder  ratio  of  7.2 : 1.0.  Thurston, 
meanwhile,  experimented  in  the  same  manner  as  Rock  wood,  determining, 
in  addition,  the  economy  of  the  high-pressure  and  intermediate  cylinders 
when  run  together  as  a  compound.  There  were  thus  two  compounds  of 
ratios  3.1 : 1  and  7.13 : 1  and  a  triple  of  ratio  1  :  3.1 :  2.3,  available  for  test. 
The  results  showed  the  7.1  compound  to  be  much  better  than  the  3.1,  but 
less  economical  than  the  triple  (24).  As  the  ratio  of  expansion  decreased, 
the  economy  of  the  intermediate  compound  closely  approached  that  of  the 
triple ;  and  at  a  very  low  ratio  it  would  probably  have  equaled  it.  The 
deduction  is  that  the  triple  engine  shows  the  efficiency  to  be  expected 
when  the  ratio  of  expansion  is  high,  as  it  should  be  for  a  triple  engine, 
but  that  a  high  ratio  ("  intermediate  ")  compound  may  far  surpass  an  ordi- 
nary compound  in  economy.  Ordinary  compound  engines  usually  have 
the  high-pressure  cylinders  too  large,  a  result  of  the  aim  toward  excessive 
overload  capacity. 

481.  Reheating.  A  considerable  gain  in  economy  is  attained  by 
superheating  the  steam  during  its  passage  through  the  receiver,  by 
means  of  pipe  coils  supplied  with  high-pressure  steam  from  the  boiler. 
The  argument  in  favor  of  reheating  is  the  same  as  that  for  the  use  of 
superheat  in  any  cylinder  (Art.  442).  It  is  not  surprising,  therefore,  that 


IV 


WITH  REHEATERS 


WITHOUT  REHEATERS 


FIGS.  215  and  216.    Art.  481.  —  Effect  of  Reheating. 

the  use  of  reheaters  is  only  profitable  when  a  considerable  amount  of 
superheating — not  less  than  100°  F. — is  effected.  Reheating  was  formerly 
unpopular,  probably  because  of  the  difficulty  of  securing  a  sufficient 
amount  of  superheat  when  saturated  steam  was  used  in  the  receiver 
coils.  With  superheated  steam,  this  difficulty  is  obviated.  Reheating 
greatly  increases  the  capacity  as  well  as  the  economy  of  the  cylinders, 
as  is  shown  by  Figs.  215  and  216,  representing  the  PV  and  TN  diagrams 
of  a  760-hp.  cross-compound  engine  (25). 

482.    Superheat  and  Jackets.     Since  multiple  expansion  itself  decreases 
cylinder  condensation,  these  refinements  cannot  be  expected  to  lead  to  such 


288 


APPLIED  THERMODYNAMICS 


large  economies  as  in  simple  engines.  Moderately  superheated  steam  has, 
however,  given  excellent  results,  eliminating  cylinder  condensation  so  per- 
fectly as  to  permit  of  wide  ranges  of  expansion  without  loss  of  economy 
and  thus  making  the  efficiency  of  the  engine,  within  reasonable  limits, 
almost  independent  of  its  load.  With  efficient  superheating  and  reheat- 
ing, jackets  are  of  little  value. 

483,  Binary  Vapor  Engine.  This  was  originated  by  Du  Tremblay  in  1850 
(26).  The  exhaust  steam  from  a  cylinder  passed  through  a  vessel  containing 
coils  filled  with  ether.  The  steam  being  at  a  temperature  of  almost  250°  F., 
while  the  atmospheric  boiling  point  of  ether  is  94°  F.,  the  latter  was  rapidly 
vaporized  at  a  considerable  pressure,  and  was  then  used  for  performing  work  in 
a  second  cylinder.  Assuming  the  initial  temperature  of  the  steam  to  have  been 
320°  F.,  and  the  final  temperature  of  the  ether  100°  F.,  the  ideal  efficiency  should 
thus  be  increased  from 


320  +  460 


320  +  460 


a  gain  of  over  200  per  cent.  The  advantage  of  the  binary  vapor  principle  arises 
from  the  low  boiling  point  of  the  binary  fluid.  This  permits  of  a  lower  tempera- 
ture of  heat  emission  than  is  possible  with  water.  Binary  engines  must  be  run 
condensing.  Since  condensing  water  is  generally  not  available  at  temperatures 
below  60°  or  70°  F.,  the  fluid  should  be  one  which  may  be  condensed  at  these  tem- 
peratures. Ether  satisfies  this  requirement,  and  gives,  at  its  initial  temperature 
of,  say,  250°  F.,  a  working  pressure  not  far  from  150  Ib.  On  account  of  its  high 
boiling  point,  however,  its  pressure  is  less  than  that  of  the  atmosphere  at  70°  F., 
and  an  air  pump  is  necessary  to  discharge  the  condensed  vapor  from  the  condenser 
just  as  is  the  case  with  condensing  steam  engines.  Sulphur  dioxide  has  a  much 
lower  boiling  point,  and  may  be  used  without  an  air 
pump :  but  its  pressure  at  250°  would  be  excessive,  and 
the  best  results  are  secured  by  allowing  the  steam  cylinder 
to  run  condensing  at  a  final  temperature  as  low  as  pos- 
sible;  at  104°  F.,  the  pressure  of  sulphur  dioxide  is  only 
90.3  Ib.  The  best  steam  engines  have  about  this  lower 
temperature  limit;  the  maximum  gain  due  to  the  use  of  a 
binary  fluid  cannot  exceed  that  corresponding  to  a  reduc- 
tion of  this  temperature  to  about  60°  or  70°  F.,  the  usual 
temperature  of  the  available  supply  of  cooling  water. 

The  steam-ether  engines  of  the  vessel  Bresil  operated 
at  43.2  Ib.  boiler  pressure  and  7.6  Ib.  back  pressure  of 
ether.  The  cylinders  were  of  equal  size,  and  the  mean 
effective  pressures  were  11.6  and  7. 1  Ib.  respectively.  The 
coal  consumption  was  brought  down  to  2.44  Ib.  per 
Ihp.-hr. ;  a  less  favorable  result  than  that  obtainable  from 
good  steam  engines  of  that  time.  Several  attempts  have 
been  made  to  revive  the  binary  vapor  engine  on  a  small  scale;  the  most  important 
recent  experiments  are  those  of  Josse  (27),  on  a  200-hp.  engine  using  steam 


Pia.217.     Art. 
59.  —  Binary  Vapor  En- 
gine. 


THE  INDICATOR 


289 


at  160  Ib.  pressure  and  200°  of  superheat,  including  three  cylinders.  The  first 
cylinders  constitute  an  ordinary  compound-condensing  steam  engine,  a  vacuum 
of  20  to  25  in.  of  mercury  being  maintained  in  the  low-pressure  cylinder  by  the 
circulation  of  sulphur  dioxide  in  the  coils  of  a  surface  condenser.  The  dioxide 
then  enters  the  third  cylinder  at  from  120  to  180  Ib.  pressure  and  leaves  it  at 
about  35  Ib.  pressure.  The  best  result  obtained  gave  a  consumption  of  167 
B.  t.  u.  per  Ihp.  per  minute,  a  result  scarcely  if  ever  equaled  by  a  high-grade 
steam  engine  (Art.  550).  The  ideal  entropy  cycle  for  this  engine  is  shown  in 
Fig.  217,  the  two  steam  cylinders  being  treated  as  one.  The  steam  diagram  is 
abcde,  and  the  heat  delivered  to  the  sulphur  dioxide  vaporizer  is  aerm.  This 
heated  the  binary  liquid  along  hi  and  vaporized  it  along  if,  giving  the  work  area 
hifg.  The  different  liquid  lines  and  saturation  curves  of  the  two  vapors  should  be 
noted.  The  binary  vapor  principle  has  been  suggested  as  applicable  to  gas  en- 
gines, in  which  the  temperature  of  the  exhaust  may  exceed  1000°  F. 

« 

ENGINE  TESTS 

484.  The  Indicator.  Two  special  instruments  are  of  prime  importance  in 
measuring  the  performance  of  an  engine.  The  first  of  these  is  the  indicator,  one 
of  the  secret  inventions  of  Watt  (28),  which 
shows  the  action  of  the  steam  in  the  cylinder. 
Some  conception  of  the  influence  of  this  device 
on  progress  in  economical  engine  operation  may 
be  formed  from  the  typically  bad  and  good  dia- 
grams of  Fig.  218.  The  indicator  furnishes  a 
method  for  computing  the  mean  effective  pres- 
sure and  the  horse  power  of  any  cylinder. 

Figure  219  shows  one  of  the  many  common 
forms.  Steam  is  admitted  from  the  engine  cylin- 
der through  6  to  the  lower  side  of  the  movable 
piston  8.  The  fluctuations  of  pressure  in  the 


Arts.  484,  486.  — Good 
and  Bad  Indicator  Diagrams. 


cylinder  cause  this  piston  to  rise  or  fall  to  an  extent  determined  by  the  stiffness 
of  the  accurately  calibrated  spring  above  it.  The  piston  movements  are  trans- 
mitted through  the  rod  10  and  the  parallel  motion  linkage  shown  to  the  pencil 
23,  where  a  perfectly  vertical  movement  is  produced,  in  definite  proportion  to 
the  movement  of  the  piston  8.  By  means  of  a  cord  passing  over  the  sheaves 
37,  27,  a  to-and-fro  movement  is  communicated  from  the  crosshead  of  the  engine 
to  the  drum,  24.  The  movements  of  the  drum,  under  control  of  the  spring,  31, 
are  made  just  proportional  to  those  of  the  piston;  so  that  the  coordinates  of  the 
diagram  traced  by  the  pencil  on  the  paper  are  pressures  and  piston  movements. 


485.  Special  Types.  Various  modifications  are  made  for  special  applications. 
For  gas  engines,  smaller  pistons  are  used  on  account  of  the  high  pressures ;  springs 
of  various  stiffnesses  and  pistons  of  various  areas  are  employed  to  permit  of  accu- 
rately studying  the  action  at  different  parts  of  the  cycle,  safety  stops  being  pro- 
vided in  connection  with  the  lighter  springs.  The  Mathot  instrument,  for 
example,  gives  a  continuous  record  of  the  ignition  lines  only  of  a  series  of  sue- 


290 


APPLIED  THERMODYNAMICS 


cessive  gas  engine  diagrams.  "  Outside-spring  "  indicators  are  a  recent  type,  in 
which  the  spring  is  kept  away  from  the  hot  steam.  The  Ripper  mean-pressure 
indicator  (29)  is  a  device  which  shows  continuously  the  mean  effective  pressure 
in  the  cylinder.  Instruments  are  often  provided  with  pneumatic  or  electrical 
operating  mechanisms,  permitting  one  observer  to  take  exactly  simultaneous  dia- 
grams from  two  or  more  cylinders.  Indicators  for  ammonia  compressors  must 
have  all  internal  parts  of  steel;  special  forms  are  also  constructed  for  heavy  hy- 


FIG.  219.    Art.  484.  —  Crosby  Steam  Engine  Indicator. 

draulic  and  ordnance  pressure  measurements.  For  very  high  speeds,  in  which  the 
inertia  of  the  moving  parts  would  distort  the  diagram,  optical  indicators  are  used. 
These  comprise  a  small  mirror  which  is  moved  about  one  axis  by  the  pressure  and 
about  another  by  the  piston  movement.  The  path  of  the  beam  of  light  is  pre- 
served by  photographing  it.  Indicator  practice  constitutes  an  art  in  itself ;  for 
the  detailed  study  of  the  subject,  with  the  influence  of  drum  reducing  motions, 
methods  of  calibration,  adjustment,  piping,  etc.,  reference  should  be  made  to  such 
works  as  those  of  Carpenter  (30)  or  Low  (31).  In  general,  the  height  of  the  dia- 
gram is  made  of  a  convenient  dimension  by  varying  the  spring  to  suit  the  maxi- 
mum pressure ;  and  accuracy  depends  upon  a  just  proportion  between  (a)  the 
movements  of  the  drum  and  the  engine  piston  and  (i)  the  movement  of  the  indi- 
cator piston  and  the  fluctuations  in  steam  pressure. 


INDICATOR  DIAGRAMS 


291 


486.  Measurement  of  Mean  Effective  Pressure.     This  may  be   accomplished 
by  averaging  a  large  number  of  equidistant  ordinates  across  the  diagram,  or, 
mechanically,  by  the  use  of  the  planimeter  (32).    In  usual  practice,  the  indicator  is 
either  piped,  with  intervening  valves,  to  both  ends  of  the  cylinder,  in  which  case  a 
pair  of  diagrams  is  obtained,  as  in  Fig.  218,  one  cycle  after  the  other,  representing 
the  action  on  each  side  of  the  piston ;  or  two  diagrams  are  obtained  by  separate 
indicators.    In  order  that  the  diagrams  may  be  complete,  the  lines  ab,  representing 
the  boiler  pressure,  cd,  of  atmospheric  pressure,  and  ef  of  vacuum  in  the  condenser, 
should  be  drawn,  together  with  the  line  of  zero  volume  ea,  determined  by  measur- 
ing the  clearance,  and  the  hyperbolic  curve  (/,  constructed  as  in  Art.  92.     The 
saturation  curve  gh  for  the  amount  of  steam  actually  in  the  cylinder  is  sometimes 
added. 

487.  Deductions.     By  taking  a  "  full-load  "  card,  and  then  one  with  the  ex- 
ternal load  wholly  removed,  the  engine  overcoming  its  own  frictional  resistance 
only,  we  at  once  find  the  me- 
chanical efficiency,  the  ratio  of     (\  s     a  \ 

power  exerted  at  the  shaft  to 
power  developed  in  the  cylin- 
der; it  is  the  quotient  of  the 
difference  of  the  two  diagrams 
by  the  former.  By  measur- 
ing the  pressure  and  the  vol- 
ume of  the  steam  at  release, 
and  deducting  the  steam  pres- 
ent during  compression,  we 
may  in  a  rough  way  com- 
pute the  steam  consumption 
per  Ihp.-hr.,  on  the  assumption 
that  the  steam  is  at  this  point 
dry ;  and,  as  in  Art.  500,  by 
properly  estimating  the  per- 
centage of  wetness,  we  may 
closely  approximate  the  actual 
steam  consumption. 

Some  of  the  applications 
of  the  indicator  are  suggested 
by  the  diagrams  of  Fig.  220. 
In  a,  we  have  admission  oc- 
curing  too  early;  in  b,  too 
late.  Excessively  early  cut-off 
is  shown  in  c ;  late  cut-off,  with 
excessive  terminal  drop,  in  d. 
Figure  e  indicates  too  early 
release ;  the  dotted  curve 
would  give  a  larger  work  area ; 

in  /,  release  is  late.    The  bad  effect  of  early  compression  is  indicated  in  g ;  late  com- 
pression gives  a  card  like  that  of  /*,  usually  causing  noisiness.    Figure  i  shows  exces- 


>'    FIG.   220. 


Art.  487.  — Indicator  Diagrams   and   Valve 
Adjustment. 


292  APPLIED  THERMODYNAMICS 

sive  throttling  during  admission;  j  indicates  excessive  resistance  during  exhaust 
which  may  be  due  to  throttling  or  to  a  poor  vacuum.  The  effect  of  a  small  supply 
pipe  is  shown  in  k,  in  which  the  upper  line  represents  a  diagram  taken  with  the 
indicator  connected  to  the  steam  chest.  The  abrupt  rise  of  pressure  along  BC  is 
due  to  the  cutting  off  of  the  flow  of  steam  from  the  steam  chest  to  the  cylinder. 
Figure  I  shows  the  form  of  card  taken  when  the  drum  is  made  to  derive  its  mo- 
tion from  the  eccentric  instead  of  the  crosshead.  This  is  often  done  in  order  to 
study  more  accurately  the  conditions  near  the  end  of  the  stroke  when  the  piston 
moves  very  slowly,  while  the  eccentric  moves  more  rapidly.  Figure  m  is  the  cor- 
responding ordinary  diagram,  and  the  two  diagrams  are  correspondingly  lettered. 
Figure  n  is  an  excellent  card  from  an  air  compressor;  o  shows  a  card  from  an  air 
pump  with  excessive  port  friction,  particularly  on  the  suction  side.  Figure  p 
shows  what  is  called  a  stroke  card,  the  dotted  line  representing  net  pressures  on 
the  piston,  obtained  by  subtracting  the  back  pressure  as  at  ab  from  the  initial 
pressure  ac,  i.e.  by  making  dc  =  ab.  Figure  q  shows  the  effect  of  varying  the 
point  of  cut-off ;  r,  that  of  throttling  the  supply.  Negative  loops  like  that  of  g 
must  be  deducted  from  the  remainder  of  the  diagram  in  estimating  the  work  done. 

488.  Measurement  of  Steam  Quality.  The  second  special  instrument  used  in 
engine  testing  is  the  steam  calorimeter,  so  called  because  it  determines  the  percent- 
age of  dryness  of  steam  by  a  series  of  heat  measurements.  Carpenter  (33)  classi- 
fies steam  calorimeters  as  follows  : 

Calorimeters 

( Barrel  or  tank 


(a)  Condensing 


1  Continuous 


f  Barrus — Continuous 


Surface  j  Hoadley 
I  Kent 

(6)  Superheating   f  External -Barrus 
I  Internal  —  Peabody 

/  N    -r^.  f  Separator 

(c)   Direct  ] '     F    .     . 

[  Chemical 

489.  Barrel  or  Tank  Calorimeter.  The  steam  is  discharged  directly  into  an 
insulated  tank  containing  cold  water.  Let  W,  w  be  the  weights  of  steam  and 
water  respectively,  t,  t\  the  initial  and  final  temperatures  of  the  water,  correspond- 
ing to  the  heat  quantities  h,  hi ;  and  let  the  steam  pressure  be  PO,  corresponding 
to  the  latent  heat  LQ  and  heat  of  liquid  ^o,  the  percentage  of  dryness  being  XQ. 
The  heat  lost  by  the  steam  is  equal  to  the  heat  gained  by  the  water ;  or,  the  steam  and 
water  attaining  the  same  final  temperature, 

W(x0L0  +  *o  -  *i)  -  t*(Ai  -  A),  whence  x»  =  h«w  +  W)-wh-Wh0 . 

WLo 

The  value  of  W  is  determined  by  weighing  the  water  before  and  after  the  mix- 
ture.    The  radiation  corrections  are  large,  and  any  slight  error  in  the  value  of  W 


CALORIMETERS 


293 


greatly  changes  the  result;  this  form  of  calorimeter  is  therefore  seldom  used,  its 
average  error  even  under  the  best  conditions  ranging  from  2  to  4  per  cent.  Some 
improvement  is  possible  by  causing  condensation  to  become  continuous  and  tak- 
ing the  weights  and  temperatures  at  frequent  intervals,  as  in  the  "Injector"  or 
"  Jet  Continuous  "  calorimeter. 

490.  Surface-condensing  Calorimeter.  The  steam  is  in  this  case  condensed 
in  a  coil ;  it  does  not  mingle  with  the  water.  Let  the  final  temperature  of  the 
steam  be  t«,  its  heat  contents  A2 ;  then 


W(xQL0 


—  A)  and  XQ  — 


-^  -  Wh0 


\V  LiQ 


More  accurate  measurement  of  W  is  possible  with  this  arrangement.  In  the 
Hoadley  form  (34)  a  propeller  wheel  was  used  to  agitate  the  water  about  the  coils; 
in  the  Kent  instrument,  arrangement  was  made  for  removing  the  coil  to  permit 
of  more  accurately  determining  IF.  In  that  of  Barrus,  the  flow  was  continuous 
and  a  series  of  observations  could  be  made  at  short  intervals. 

491.  Superheating  Calorimeters.  The  Peabody  throttling  calorimeter 
is  shown  in  Fig.  221 ;  steam  entering  at  b  through  a  partially  closed  valve 
expands  to  a  lower  steady  pressure  in  A  and  then  flows  into  the  atmos- 
phere. Let  LQ,  h0,  XQ  be  the  condition  at  b,  and  assume  the  steam  to  be 
superheated  at  A,  its  temperature  being  T,  t  being  the  6 

temperature  corresponding  to  the  pressure  p,  and  the  cor- 
responding total  heat  at  saturation  H.  Then,  the  total  heat 
at  b  equals  the  total  heat  at  A,  or 


where  k  is  the  mean  specific  heat  of  superheated  steam 
at  the  pressure  p  between  T  and  t ;  whence 

_H+k(T-t)-h0 
~Lo~ 

If  we  assume  the  pressure  in  A  to  be  that  of  the  atmos- 
phere, H=  1150.4,  and  superheating  is  possible  only  when 
x0LQ  +  ho  exceeds  1150.4.  For  each  initial  pressure,  then, 
there  is  a  corresponding  minimum  value  of  x0  beyond 
which  measurements  are  impossible;  thus,  for  200  lb., 
7,o  =  843.2,fto  =  354.9,  and  XQ  (minimum)  is  0.94.  Aside 
from  this  limitation,  the  throttling  calorimeter  is  exceed- 
ingly accurate  if  the  proper  calibrations,  corrections,  and  methods  of 
sampling  are  adopted.  In  the  Barrus  throttling  calorimeter,  the  valve  at 
b  is  replaced  by  a  diaphragm  through  which  a  fine  hole  is  drilled,  and  the 
range  of  x0  values  is  increased  by  mechanically  separating  some  of  the 
moisture.  The  same  advantage  is  realized  in  the  Barrus  superheating 
calorimeter  by  initially  and  externally  heating  the  sample  of  steam.  The 


FIG.  221.  Art.  491. 
—  Superheat- 
ing Calorimeter. 


294 


APPLIED  THERMODYNAMICS 


amount  of  heat  thus  used  is  applied  in  such  a  way  that  it  may  be  ac- 
curately measured.     Let  it  be  called,  say,  Q  per  pound.     Then 

x0L0  +  h0  +  Q  =  H+  k(T—  t),  and  XQ  -- 


492.  Separating  Calorimeters.  The  water  and  steam  are  mechanically  sepa- 
rated and  separately  weighed.  In  Fig.  222,  steam  enters,  through  6,  the  jacketed 
chamber  shown.  The  water  is  intercepted  by  the  cup 
14,  the  steam  reversing  its  direction  of  flow  at  this 
point  and  entering  the  jacket  space  7,  4,  whence  it  is 
discharged  through  the  small  orifice  8.  The  water  ac- 
cumulates in  3,  its  quantity  being  indicated  by  the 
gauge  glass  10.  The  quantity  of  steam  flowing  is  de- 
termined by  calibration  for  each  reading  of  the  gauge 
at  9.  The  instrument  is  said  to  be  fairly  accurate  un- 
less the  percentage  of  moisture  is  veiy  small.  The 
steam  may  be,  of  course,  run  off,  condensed,  and 
actually  weighed. 

493-  Chemical  Calorimeter.     This   depends  for  its 
action  on  the  fact  that  water  will  dissolve  certain  salts 

.(e.g.   sodium   chloride)    which   are   insoluble    in    dry 
steam. 

494-  Electric  Calorimeter.     The  Thomas  superheat- 
ing and  throttling  instrument  (35)  consists  of  a  small 
soapstone   cylinder   in  which  are  embedded  coils   of 

FIG  222     Art  492  —  S          German   silver  wire,   constituting   an  electric   heater, 
rating  Calorimeter.  ^is  *s  inserted  in  a  brass  case  through  which  flows 

a  current  of  steam.  The  electrical  energy  correspond- 
ing to  heat-augmentation  to  any  superheated  condition  being  known,  say,  as 
E  B.  t.u.  per  pound  (1  B.  t.  u.  =  17.59  watts  per  minute),  we  have,  as  in  Art.  491, 

x0L0  +  7>0  +  E  =  II  +  k(T  -  0,  whence  x0  =  H  +  k(T  -  t}-\- E  ^ 

Lo 

495.  Engine  Trials:  Heat  Measurement.  We  may  ascertain  the  heat 
supplied  in  the  steam  engine  cycle  either  by  direct  measurement,  or  by 
adding  the  heat  equivalent  of  the  external  work  done  to  the  measured  amount 
of  heat  rejected.  In  the  former  case  the  amount  of  water  fed  to  the  boiler 
must  be  determined,  by  weighing,  measuring,  or  (in  approximate  work)  by 
the  use  of  a  water  meter.  The  heat  absorbed  per  pound  of  steam  is  ascer- 
tained from  its  temperature,  quality,  and  pressure,  and  the  temperature  of 
the  water  fed  to  the  boiler.  In  the  latter  case,  the  steam  leaving  the 
engine  is  condensed  and,  in  small  engines,  weighed ;  or  in  larger  engines, 
determined  by  metering  or  by  passing  it  over  a  weir.  This  latter  of  the 
two  methods  of  testing  has  the  advantage  with  small  engines  of  greater 


ENGINE  TEST 


295 


accuracy  and  of  giving  accurate  results  in  a  test  of  shorter  duration.  Where 
the  engine  is  designed  to  operate  non-condensing,  the  steam  may  be  con- 
densed for  the  purposes  of  the  test  by  it  passing  it  over  coils  exposed  to 
the  atmosphere,  so  that  no  vacuum  is  produced  by  the  condensation.  If 
jackets  are  used,  the  condensed  steam  from  them  must  be  trapped  off  and 
weighed.  This  water  would  ordinarily  boil  away  when  discharged  at 
atmospheric  pressure,  so  that  provision  must  be  made  for  first  cooling  it. 

496.  Heat  Balance.     By  measuring  loth  the  heat  supplied  and  that  rejected,  as 
well  as  the  work  done,  it  is  possible  to  draw  up  a  debit  and  credit  account  show- 
ing the  use  made  of  the  heat  and  the  unaccounted  for  losses.     These  last  are  due 
to  the  discharge  of  water  vapor  by  the  air  pump,  to  radiation,  and  to  leakage. 
The  weight  of  steam  condensed  may  easily  be  four  or  five  per  cent  less  than  that 
of  the  water  fed  to  the  boiler.     Let  77,  h,  be  the  heat  contents  of  the  steam  and 
the  heat  in  the  boiler  feed  water  respectively;  the  heat  absorbed  per  pound  is 
then  H  —  h.     Let  Q  be  the  heat   contents  of   the  exhausted   steam    (measured 
above  the  feed  water  temperature)  and   W  the  heat  equivalent  of  the  work  done 
per  pound.     Then  for  a  perfect  heat  balance,  H  —  h  =  Q  +  W.     In  practice,  W 
is  directly  computed  from  the  indicator  diagrams ;  H  and  Q  must  be  corrected 
for  the  quality  of  steam  as  determined  by  the  calorimeter  or  otherwise. 

* 

497.  Checks ;  Codes.     Where  engines  are  used  to  drive  electrical  generators 
the  measurement  of  the  electrical  energy  gives  a  close  check  on  the  computation 
of  indicated  horse  power.     In  locomotive  trials  a  similar  check  is  obtained  by 
comparison  of  the  drawbar  pull  and  speed  (36).      In  turbines,  the  indicator 
cannot  be  employed;  measurement  of  the  mechanical  power  exerted  at  the  shaft 
is  effected  by  the  use  of  the  friction  brake.     Standard  codes  for  the  testing  of 
pumping  engines  (37),  and  of  steam  engines  generally  (38),  have  been  developed 
by  the  American  Society  of  Mechanical  Engineers. 

498.  Example  of  an  Engine  Test.     Figure  224,  from  Hall  (39),  gives 
the  indicator  diagrams  from  a  30  and  56  by  72-in.  compound  engine  at 
58  r.  p.  m.     The  piston  rods  were  4J  and  5J  in.  diameter.     The  boiler 


FIG.  224.     Arts.  498,  499,  500.  — Indicator  Cards  from  Compound  Engine. 


296  APPLIED  THERMODYNAMICS 

pressure  was  124.0  Ib.  gauge:  the  pressure  in  the  steam  pipe  near  the 
engine,  122.0  Ib.  The  temperature  of  jacket  discharge  was  338°  F.  The 
conditions  during  the  calorimetric  test  of  the  inlet  steam  were  PQ  =  122.08 
Ib.  gauge,  T  —  302.1°  F.  (Art.  491),  pressure  in  calorimeter  body  (Fig.  221), 
11.36  Ib.  (gauge).  The  net  weight  of  boiler  feed  water  in  12  hours  was 
231,861.7  Ib. ;  the  weight  of  water  drained  from  the  jackets,  15,369.7  Ib. 

From  the  cards,  the  mean  effective  pressures  were  44.26  and  13.295 
Ib.  respectively;  and  as  the  average  net  piston  areas  were  697.53  and 
2452.19  square  inches  respectively,  the  total  piston  pressures  were  44.26 
X  697.53=30872.7  and  13.295  x  2452.19=32601.9  Ib.  respectively.  These 
were  applied  through  a  distance  of 

if  x  2  x  58  =  696  feet  per  minute ; 
whence  the  indicated  horse  power  was 

(30872.7  +  32601.9)  x  696  = 
33000 

From  Art.  491,  xGL0-^-h()  =  H-{-k  (T-  t),  or  in  this  case,  866.5 aj  +  322.47 
=  1155.84  +  0.48*  (302.1-242.3)  whence  x0  =  0.995.  The  weight  of 
cylinder  feed  was  231,861.7-15,369.7  =  216,492.0  Ib.  At  its  pressure  of 
136.7  Ib.  absolute,  £=866.5,  ft  =  322.4.  For  the  ascertained  dryness,  the 
total  heat  per  pound,  above  32°,  is  322.4  +  (0.995  x  866.5)  =  1184.5  B.  t.  u. 
The  heat  left  in  the  steam  at  discharge  from  the  condenser  (at  114°  F.) 
was  82  B.  t.  u. ;  the  net  heat  absorbed  per  pound  of  cylinder  feed  was 
then  1184.5-82.0  =  1102.5;  for  the  total  weight  of  cylinder  feed  it  was 
1102.5  x  216,492  =  238,682,430  B.  t.  u.  The  total  heat  in  one  pound  of 
jacket  steam  was  also  1184.5  B.  t.  u.  This  was  discharged  at  338°  F. 
(7i  =  308.8),  whence  the.  heat  utilized  in  the  jackets  was  1184.5  —  308.8 
=  875.7  B.  t.  u.  (The  heat  discharged  from  both  jackets  and  cylinders 
was  transferred  to  the  boiler  feed  water,  the  former  at  338°,  the  latter  at 
114°  F.)  The  supply  of  heat  to  the  jackets  was  then  875.7  x  15,369.7 
=13,459,246.29  B.  t.  u :  the  total  to  cylinders  and  jackets  was  this  quan- 
tity plus  238,682,430  B.  t.  u.,  or  252,141,676.29  B.  t.  u.  Dividing  this  by 
60  X 12  we  have  350,196.77  B.  t.  u.  supplied  per  minute. 

499.     Statement  of  Results.     We  have  the  following : 

(a)  Pounds   of   steam  per   Ihp.-hr.  =  231,861.7  -=-  12  -=-  1338.62  =  14.43. 
•  (This  is  the  most  common  measure  of  efficiency,  but  is  wholly 
unsatisfactory  when  superheated  steam  is  used.) 

*  Value  taken  for  the  specific  heat  of  superheated  steam. 


STEAM   CONSUMPTION   FROM   DIAGRAM  297 

(6)  Pounds  of  dry  steam  per  Ihp.-hr.  =  14.43  x  0.995  *  =  14.36. 

(c)  Heat  consumed  per  Ihp.  per  minute  =  350,196.77  -^  1338.62  =  261.61 

B.  t.  u. 

(d)  Thermal  efficiency  =  ^fooo  ^261.61  =  0.1621. 

(e)  Work  per  pound  of  steam=  25^14^67p9  X  0.1621  =  176  B.  t.  u. 


(g)  Clausius  efficiency  (Art.  409),  with  dry  steam, 

810.82 

•6  log*   573.6  __ 
351.22-114-f866.5~ 

(h)  Ratio  of  (d)  -=-  (g)  =  0.1621  -5-  0.265  =  0.61. 

500.  Steam  Consumption  from  Diagram.  The  inaccuracy  of  such  estimates 
will  be  shown.  In  the  high-pressure  cards,  Fig.  224,  the  clearance  space  at  each 
end  of  the  cylinder  was  0.932  cu.  ft.  The  piston  displacement  per  stroke  on  the  side 
opposite  the  rod  was  706.86  x  72  -4-  1728  =  29.453  cu.  ft. ;  the  cylinder  volume 
on  this  side  was  29.453  +  0.932  =  30.385  cu.  ft.  The  length  of  the  correspond- 
ing card  (a)  is  3.79  in. ;  the  clearance  line  be  is  then  drawn  distant  from  the 
admission  line 


3.79  x  .  =  0.117  in. 

29.453 

At  d,  on  the  release  line,  the  volume  of  steam  is  30.385  cu.  ft.,  and  its  pressure  is 
31.2  Ib.  absolute.  From  the  steam  table,  the  weight  of  a  cubic  foot  of  steam  at 
this  pressure  is  0.076362  Ib. ;  whence  the  weight  of  steam  present,  assumed  dry,  is 
0.076362  x  30.385  =  2.3203  Ib.  At  a  point  just  after  the  beginning  of  compres- 
sion, point  e,  the  volume  of  steam  expressed  as  a  fraction  of  the  stroke  plus  the 
clearance  equivalent  is  0.517  -r-  3.907  =  0.1321,  3.907  being  the  length  bff  in  inches. 
The  actual  volume  of  steam  at  e  is  then  0.1321  x  30.385  =  4.038  cu.  ft.,  and  its 
pressure  is  28.3  Ib.  absolute,  at  which  the  specific  weight  is  0.069683  Ib.  The 
weight  present  at  e  is  then  4.038  x  0.069683  =  0.280  Ib.  The  net  weight  of  steam 
used  per  stroke  is  2.3203  -  0.280  =  2.0403  Ib.,  or,  per  hour,  2.0403  x  58  x  60  =  7090 
Ib.,  for  this  end  of  the  cylinder  only.  For  the  other  end,  the  weight,  similarly 
obtained,  is  7050  Ib. ;  the  total  weight  is  then  14,140  Ib.  The  horse  power 
developed  in  the  high-pressure  cylinder  is  650,  and  the  cylinder  feed  per  Ihp.-hr. 
from  high-pressure  diagrams  is  21.8  Ib.  The  same  process  may  be  applied  to  the 
low-pressure  diagrams.  It  is  best  to  take  the  points  d  and  e  just  before  the  begin- 
ning of  release  and  after  the  beginning  of  compression  respectively.  The  method 

*The  factor  0.995  does  not  precisely  measure  the  ratio  of  energy  in  the  actual 
steam  to  that  in  the  corresponding  weight  of  dry  steam,  but  the  correction  is  usually 
made  in  this  way. 


298  APPLIED  THERMODYNAMICS 

is  widely  approximate,  but  may  give  results  of  some  value  in  the  absence  of  a 
standard  trial,  if  the  quality  of  steam  at  release  and  compression  is  known  (Arts. 
448,  440). 

501  .  General  Analysis.  Let  A  ,  a  represent  the  areas  of  the  two  sides  of  the 
piston,  P,  p  the  corresponding  mean  effective  pressures,  S  the  length  of  the  stroke, 
and  R  the  number  of  revolutions  per  minute.  The  indicated  horse  power  is,  then, 

(AP  +  ap)SR 


33000 

Let  B,  I  denote  the  ratios  of  volume  at  release  to  total  cylinder  volume,  W,  wt 
the  corresponding  specific  weights,  T,  t  the  ratios  of  volume  at  compression  to 
total  cylinder  volume,  and  F,  v  the  corresponding  specific  weights  at  that  point  ; 
then,  if  C,  c  denote  the  clearance  volumes,  the  volumes  of  steam  at  release  are 
B(AS  +  C)  and  b(aS  +  c)  ;  the  weights  are  WB(AS  +  C)  and  wb(aS  +  c)  ;  the 
volumes  at  compression  are  T(A  S  +  C)  and  t(aS  -f  c)  ;  the  weights  there  are 
VT(AS  +  C)  and  vt(aS  +  c)  ;  the  weight  of  cylinder  feed  per  revolution  is  then 
WB(AS  +  C)  +  wb(aS  +  c)  -  VT(AS  +  C)  -  vt(aS  +  c)  ;  or,  per  hour,  it  is  60  R 
times  this.  The  quotient  of  this  expression  by  that  given  for  horse  power  gives 
the  steam  consumption  per  indicated  horse  power  hour,  directly  derived  from  the 
cards;  and  if  C,  c  be  expressed  as  functions  of  the  area  and  stroke,  say  as  aAS, 
in  which  a  is  the  ratio  of  clearance  to  piston  displacement,  the  values  of  A,  S,  and  R 
cancel  out  so  that  no  information  is  necessary  other  than  that  given  by  the  diagrams 
themselves. 

502.  Measurement  of  Rejected  Heat.  A  common  example  is  in  tests  in 
which  the  steam  is  condensed  by  a  jet  condenser  (Art.  584).  In  a  test 
cited  by  Ewing  (40),  the  heat  absorbed  per  revolution  measured  above  the 
temperature  of  the  boiler  feed  was  1551  B.  t.  u.  ;  that  converted  into  work 
was  225  B.  t.  u.  The  exhaust  steam  was  mingled  with  the  condensing 
water,  a  combined  weight  of  51.108  Ib.  being  found  per  revolution.  The 
temperature  of  the  entering  water  was  50°  F.,  that  of  the  discharged  mix- 
ture was  73.4°  F.,  and  the  cylinder  feed  amounted  to  1.208  Ib.  per  revolu- 
tion. The  temperature  of  the  boiler  feed  water  was  59°  F.  We  may 
compute  the  injection  water  as  51.108  —  1.208  =  49.9  Ib.  and  the  heat 
absorbed  by  it  as  approximately  49.9(73.4  -  50)=  1167  B.  t.  u.  The 
1.208  Ib.  of  feed  were  discharged  at  73.4°,  whereas  the  boiler  feed  was  at 
59°  ;  a  heat  rejection  of  73.4  -  59  =  14.4°  occurred,  or  14.4  x  1.208  =  17.4 
B.  t.  u.  The  total  heat  rejection  was  then  1167  -f  17.4  =  1184.4  B.  t.  u., 
to  which  we  must  add  47  B.  t.  u.  from  the  jackets,  giving  a  total  of 
1231.4  B.  t.  u.  Adding  this  to  the  work  done,  we  have  1231.4  -f  225  = 
1456.4  B.  t.  u.  accounted  for  of  the  total  1551  B.  t.  u.  supplied;  the 
discrepancy  is  over  6  per  cent. 

When  surface  condensers  are  used,  the  temperatures  of  discharge  of 
the  condensed  steam  and  the  condenser  water  are  different  and  the  weight 


HIRN'S  ANALYSIS  299 

of  water  is  ascertained  directly.     In  other  respects  the  computation  would 
be  as  given.* 

503.  Statements  of  Efficiency.   Engines  are  sometimes  rated  on  the  basis  of 
fuel  consumption.     The  duty  is  the  number  of  foot-pounds  of  work  done  in  the 
cylinder  per  100  pounds  of  coal  burned.     The  efficiency  of  the  plant  is  the  quotient 
of  the  heat  converted  into  work  per  pound  of  coal,  by  the  heat  units  contained  in 
the  pound  of   coal.     In  the  test  in  Art.  498,  the  coal  consumption  per  Ihp.-hr. 
was  2068.84  H-  1338.62  =  1.54  Ib.     In  some  cases,  all  statements  are  based  on  the 
brake  horse  power  instead  of  the  indicated  horse  power.     The  ratio  of  the  two  is  of 
course  the  mechanical  efficiency.     It  may  be  noted  that  the  engine  is  charged  with 
steam,  not  at  boiler  pressure,  but  at  the  pressure  in  the  steam  pipe.     The  differ- 
ence between  the  two  pressures  and  qualities  represents  a  loss  which  may  be  con- 
sidered as  dependent  upon  the   transmissive  efficiency.     The  plant  efficiency   is 
obviously  the  product  of  the  efficiencies  of  boiler  (Art.  574),  transmission,  and 
engine. 

504.  Measurement  of  Heat  Transfers  :  Hirn's  Analysis.    In  the  refined  methods 
of  studying  steam  engine  performance  developed  by  Hirn  (41),  and  expounded  by 
Dwelshauvers-Dery  (42),  the  heat  absorbed       P 

and  that  rejected  are  both  measured.  Dur- 
ing any  path  of  the  cycle,  the  heat  inter- 
change between  fluid  and  walls  is.  computed 
from  the  change  in  internal  energy,  the  heat 
externally  supplied  or  discharged,  and  the 
external  work  done. 

In  Fig.  225,  consider  the  cycle  as  made 
up  of  the  four  paths,  01,  12,  23,  30,  called 
respectively  a,  6,  c,  d.     Let  M,  represent  the       FlG.  225.    Art.  504.  _  Hirn's  Analysis. 
weight  of   cushion  steam,    and   M  that   of 

cylinder  feed,  per  stroke.     We  have  then  the  following  expressions  for  internal 
energy  :  _ 


The  general  equation  for  heat  transfers  is  II  =  T  +  I  +  W,  H  standing  for 
heat  supplied  or  withdrawn,  T  +  1  for  a  change  in  internal  energy,  and  W  for 
external  work  done  or  consumed.  In  order  to  avoid  confusion  in  algebraic  signs, 
we  will  regard  +  H  as  representing  a  reception  of  heat  by  the  fluid,  +  W  as 
denoting  positive  work  done  by  it,  and  +  (T+  /)(here  represented  by  the  symbol 
E  with  a  subscript)  as  specifying  a  gain  of  internal  energy.  Let  Qa,  Qb,  Qc,  Q* 
represent  amounts  of  heat  transferred  to  the  walls  along  the  paths  a,  ft,  c,  d. 

Consider  the  path  a.     Let  the  heat  supplied  by  the  incoming  steam  be   Q. 

Then  Q-Qa=  Wa+(El-E,}. 

*  It  is  most  logical  to  charge  the  engine  with  the  heat  measured  above  the  tem- 
perature of  heat  rejection.  This,  in  Fig.  182,  for  example,  makes  the  efficiency 

%,  rather  than  -  dfbc    ,  the  ordinate  YX  representing  the  feed-water  temperature. 


ideb/,  YXebZ 


300 


APPLIED  THERMODYNAMICS 


Along  the  path  &,  -  Qb  =  Wb+  (£2-  EI)  ;   along  d,  -  Qd=  -  Wd  +  (E0  -  Ei). 

Along  c,  heat  is  carried  away  by  the  discharged  steam  and  by  the  cooling 
water.  Let  G  denote  the  weight  of  cooling  water  per  stroke,  k5  and  h4  its  final  and 
initial  heat  contents,  and  h6  the  heat  contents  of  the  discharged  steam.  The  heat 
rejected  by  the  fluid  per  stroke  is  then  G(h5  —  A4)+  MhG.  Then  —  Qc  —  G(h5  —  /*4) 
-  Mh6  =  -  Wc  +  (Es  -  E2),  and  Qc  =  -  G(h5  -  7*4)  -  Mhf>  +  Wc  -  (Es  -  Ez). 

The  values  of  x  at  the  four  points  of  the  cycle  are  obtained  by  comparing  the 
volumes  at  those  points  with  the  volumes  of  saturated  steam  at  the  same  pressure. 
If  the  cushion  steam  and  cylinder  feed  per  stroke,  and  the  quality  of  the  latter  as 
supplied,  be  known,  with  the  values  of  7*4,  h5,  and  hG  and  the  weight  of  cooling 
water,  we  may  then  find  values  of  Qa,  Qb,  Qc,  and  Qd  from  the  indicator  diagram 
alone,  the  OP  axis  being  properly  located. 

505.  Graphical  Representation.     In  Fig.  226,  from  the  base  line  xy,  we  may 
lay  off  the  areas  oefs  representing  heat  lost  during  admission,  smba  showing  heat 

gained  during  expansion,  m^cr'showing  heat  gained 
during  release,  and  oakr  showing  heat  lost  during 
compression.  If  there  were  no  radiation  losses 
from  the  walls  to  the  atmosphere,  the  areas  above 
the  line  xy  would  just  equal  those  below  it.  Any 
excess  in  upper  areas  represents  radiation  losses. 
Ignoring  these  losses,  Him  found  by  comparing  the 
work  done  with  the  value  of  Q  —  Mhc>  —  G(h.  —  7*4) 
— v  an  approximate  value  for  the  mechanical  equivalent 
T  o  r\  f_  ~^~y  of  heat  (Art.  32). 

Analytically,  if  Qr  denote  the  loss  by  radiation, 
its  value   is   the    algebraic  sum  of  Qa,  Qb,  Qc,  Qd. 

'IG'    2"G'Tra^nstferiS05'~Heat       Tf  the  heat  Qj  be  suPPlied  bv  a  steam   Jacket»  then 

Qr  =  Qj  4-  2Q0,  z.,  c,  d>  The  heat  transfer  during  re- 
lease, Qc,  regarded  by  Him  as  in  a  special  sense  a  measure  of  wastefulness  of 
the  walls,  maybe  expressed  as  Qr  —  Qj  —  2Qa, &,<*•  In  a  non-condensing  engine, 
Qr  can  be  determined  only  by  direct  experiment.  In  most 
cases  the  value  of  M0  is  computed  on  the  assumption  that 
%  =  1.0  (Art.  440). 

\ 

TYPES  OF  STEAM  ENGINE 

506.  Special  Engines.     We  need  not  consider  the  com- 
mercially unimportant  class  of  engines  using  vapors  other 
than  steam,    those    experimental  engines  built   for   educa- 
tional institutions   which  belong   to  no  special  type   (4-i), 
engines    of  novel   and  limited  application  like   those   em- 
ployed on  motor  cars  (44),  nor  the  "fire less"  or  stored  hot- 
water  steam  engines  occasionally  employed  for  locomotion 
(45). 

A  novel  form  of  heat  engine,  the  puhometer,  is  shown 

in    Fig.   227.     It  is   employed   solely   for   pumping  water.     pIG  227.    Art.  50t>. 

Steam  enters  at  B,  water  at  E.     The  ball  C  being  in  the  Pulsometer. 


TYPES  OF   ENGINE 


301 


position  shown,  the  steam  forces  water  contained  in  A  through  the  check  valve  V 
into  a  delivery  passage  D.  When  the  water  level  sinks  so  far  that  steam  begins  to 
blow  through  Z),  violent  agitation  is  produced,  and  the  steam  begins  to  condense. 
The  partial  vacuum  causes  the  ball  C  to  rock  over  so  as  to  close  chamber  A,  and 
also  causes  water  to  rise  through  E  and  the  suction  check  valve  X,  again  filling  A. 
Meanwhile,  the  same  series  of  actions  has  started  in  W.  The  only  moving  parts 
are  the  ball  and  check  valves.  The  efficiency  is  usually  under  2  per  cent. 
Letting  the  heat  lost  by  the  steam  be  x0L0  +  h0—hv  that  gained  by  the  water 
being  Jiv  —  hy  the  heat  equation  for  y  pounds  of  water  pumped  per  pound  of 
steam  used  is  a:0L0  +  h0  —  hl  =  y(hl  —  fi2).  If  the  total  head  be  s,  the  work  is 
s(y  -f  1)  foot-pounds  (ignoring  the  fact  that  the  condensed  steam  is  received 
without  head),  and  the  efficiency  is  s(y  +  1)  -j-  (x0L0  -f  h0  —  Aj). 

507.  Classification  of  Engines.  Commercially  important  types  may  be  con- 
densing or  non-condensing.  They  are  classified  as  right-hand  or  left-hand,  accord- 
ing as  the  fly  wheel  is  on  the  right  or  left  side  of  the  center  line  of  the  cylinder, 
as  viewed  from  the  back  cylinder  head.  They  may  be  simple  or  multiple-expan- 


FIG.  228.     Art.  507.  —  Angle-Compound  Engine.     (American  Ball  Engine  Company.) 


302 


APPLIED  THERMODYNAMICS 


sion,  with  all  the  successive  stages  and  cylinder  arrangements  made  possible  in 
the  latter  case.  They  may  be  single-acting  or  double-acting ;  the  latter  is  the  far 
more  usual  arrangement.  They  may  be  rotative  or  non-rotative.  The  direct-acting 
pumping  engine  is  an  example  of  the  latter  type;  the  work  done  consists  in  a 
rectilinear  impulse  at  the  water  cylinders.  In  the  duplex  engine,  simple  cylinders 
are  used  side  by  side.  The  terms  horizontal,  vertical,  and  inclined  refer  to  the  posi- 
tions of  the  center  lines  of  the  cylinders.  The  horizontal  engine,  as  in  Figs.  186 
and  229,  is  mostly  used  in  land  practice ;  the  vertical  engine  is  most  common  at 


FIG.  229.     Art.  507.  —  Automatic  Engine.     (American  Ball  Engine  Company.) 

sea.  Cross-compound  vertical  engines  are  often  direct-connected  to  electric  gen- 
erators. Vertical  engines  have  occasionally  been  built  with  the  cylinder  below 
the  shaft.  This  type,  with  the  inclined  engine,  is  now  rarely  used.  Inclined 
engines  have  been  built  with  oscillating  cylinders,  the  use  of  a  crosshead  and 
connecting  rod  being  avoided  by  mounting  the  cylinder  on  trunnions,  through 
which  the  steam  was  admitted  and  exhausted.  Figure  228  shows  a  section  of 


TYPES   OF   ENGINE 


303 


the  interesting  angle-compound,  in  which  a  horizontal  high-pressure  cylinder 
exhausts  into  a  vertical  low-pressure  cylinder.  A  different  type  of  engine,  but 
with  a  similar  structural  arrangement,  has  been  used  in  some  of  the  largest 
power  stations. 

Engines  are  locomotive,  stationary,  or  marine.  The  last  belong  in  a  class  by 
themselves,  and  will  not  be  illustrated  here ;  their  capacity  ranges  up  to  that  of 
our  largest  stationary  power  plants.  Stationary  engines  are  further  classed  as 
pumping  engines,  mill  engines,  power  plant  engines,  etc.  They  may  be  further 
grouped  according  to  the  method,  of  absorbing  the  power,  as  belted,  direct-con- 
nected, rope  drive,  etc.  An  engine  directly  driving  an  air  compressor  is  shown  in 
Fig.  86.  "  Rolling  mill  engines  "  undergo  enormous 
variations  in  load,  and  must  have  a  correspondingly 
massive  (tangye)  frame.  Power  plant  engines  gen- 
erally must  be  subjected  to  heavy  load  variations; 
their  frames  are  accordingly  usually  either  tangye  or 
semi-tangye.  Mill  engines  operate  at  steadier  loads, 
and  have  frequently  been  built  with  light  girder 
frames.  Modern  high  steam  pressures  have,  however, 
led  to  the  general  discontinuance  of  this  frame  in 
favor  of  the  semi-tangye. 

A  slow-speed  engine  may  run  at  any  speed  up  to 
125  r.  p.  m.  From  125  to  200  r.  p.  m.  may  be  re- 
garded as  medium  speed.  Speeds  above  200  r.  p.  m. 
are  regarded  as  high.  Certain  types  of  engine  are 
adapted  only  for  certain  speed  ranges ;  the  ordinary 
slide-valve  engine,  shown  in  Fig.  186,  may  be  oper- 
ated at  almost  any  speed.  For  large  units,  speeds 
range  usually  from  80  to  100  r.  p.  m.  The  higher- 
speed  engines  are  considered  mechanically  less  re- 
liable, and  their  valves  do  not  lend  themselves  to  quite 
as  economical  a  distribution  of  steam.  An  important 
class  of  medium-speed  engines  has,  however,  been  in- 
troduced, in  which  the  independent  valve  action  of 
the  Corliss  type  has  been  retained,  and  the  promptness 
of  cut-off  only  attainable  by  a  releasing  gear  has  been 
approximated.  In  some  cheap  high-speed  engines 
governing  is  effected  simply  but  uneconomically  by 
throttling  the  steam  supply.  Such  engines  may  have 
shallow  continuous  frames  or  the  sub-base,  as  in  Fig. 
229,  which  represents  the  large  class  of  automatic 
high-speed  engines  in  which  regulation  is  effected  by 
automatically  varying  the  point  of  cut-off.  Figure  230 
shows  three  sets  of  indicator  diagrams  from  a  com- 
pound engine  of  this  type,  running  non-condensing 
at  various  loads.  Some  of  the  irregulations  of  these 
diagrams -are  without  doubt  due  to  indicator  inertia;  but  they  should  be  care- 
fully compared  with  those  showing  the  steam  distribution  with  a  slow-speed 


304 


APPLIED  THERMODYNAMICS 


releasing  gear,  in   Fig.  218.     All  of  the  so-called  "  automatic "  engines  run   at 
medium  or  high  rotative  speeds. 

The  throttling  engine  is  used  only  in  special  or  unimportant  applications.  The 
automatic  type  is  employed  where  the  comparatively  high  speed  is  admissible,  in 
units  of  moderate  size.  Better  distribution  is  afforded  by  the  four -valve  engine,  in 


,CoHiss  Steom  Valve 
Planished  Sheet  Steel  • 


Corliss  Exhaust Volve 


Steam  pipe 


'team  Flanqt 
Throttle  Valve 


heat  insulating  filling 
Steam  inlet 
Back  Cylinder  Head 

Bach  Cylinder  Head  Studs-' 

Bach  Cijl  Head 


Planished  Steel  Lagqmg 
Heat  insulating  Pilling 

/         Corliss  Steam Volve  Chamber 


Steam  Port 
Corevtnr 

,   Pluqs. 
Piston  Rod  Nut. 

ExhflitstPor 


Cylinder  Head  5tod$ 
Piston  Rod  Gland  Studs 
iston  Rod  Gland 


Piston  Rod 
3iston 

ExhoustPort 


w^vu  £££>££ 

S^y^v^J-V^*^^.     ^' '''^X\\\\\- 
Exhaust  Chest 


stonRod  Pochinq 


orhss  E  xhoust  Volve 
Planished  Sheet  5t««ILoqqmq 
-Heat  msulotmg  Filling 


ETfiaust  Flange 
""Exhaust  Opening 
Lirhaust  Pipe 

FIG.  231.    Art.  507.  — Corliss  Engine  Details.     (Murray  Iron  Works  Company.) 


THE   STEAM   POWER   PLANT 


305 


which  the  four  events  of  the  stroke  may  be  independently  adjusted,  and  this  type 
is  often  used  at  moderately  high  speeds.  Sharpness  of  cut-off  is  usually  obtainable 
only  with  a  releasing  gear,  in  which  the  mechanism  operating  the  valves  is  discon- 
nected, and  the  steam  valve  is  au- 
tomatically and  instantaneously 
closed.  This  feature  distinguishes 
the  Corliss  type,  most  commonly 
used  in  high-grade  mill  and  power 
plant  service.  With  the  releasing 
gear,  usual  speeds  seldom  exceed 
100  r.  p.  m.  The  valve  in  a  Cor- 
liss engine  is  cylindrical,  and  ex- 
tends across  the  cylinder.  Some 
details  of  the  mechanism  are 
shown  in  Fig.  231.  In  very  large 
engines,  the  releasing  principle  is 
sometimes  retained,  but  with 
poppet  or  other  forms  of  valve. 
Figure  232  shows  the  parts  of  a 
tvpical  Corliss  engine  with  semi- 
tangye  frame. 


508.  The  Steam  Power  Plant 
Figure  233,  from  Heck  (46),  is 
introduced  at  this  point  to  give 
a  conception  of  the  various  ele- 
ments composing,  with  the  en- 
gine, the  complete  steam  plant. 
Fuel  is  burned  on  the  grate  1 ; 
the  gases  from  the  fire  follow 
the  path  denoted  by  the  arrows, 
and  pass  the  damper  4  to  the 
chimney  5.  Water  enters, 
from  the  pump  IV,  the  boiler 
through  29,  and  is  evaporated, 
the  steam  passing  through  8  to 
the  engine.  The  exhaust  steam 
from  the  engine  goes  through 

18    to    the    condenser   III,    to  °*      \.  *    fs 

which  water  is  brought  through 

21.  Steam  to  drive  the  condenser  pump  comes  from  26.  Its  exhaust, 
with  that  of  the  feed  pump  31,  passes  to  the  condenser  through  27.  The 
condensed  steam  and  warmed  water  pass  out  through  23,  and  should,  if 
possible,  be  used  as  a  source  of  supply  for  the  boiler  feed.  The  free  ex- 
haust pipe  19  is  used  in  case  of  breakdown  at  the  condenser. 


306 


APPLIED  THERMODYNAMICS 


509.  The  Locomotive. 
This  is  an  entire  power  plant, 
made  portable.  Figure  234 
shows  a  typical  modern  form. 
The  engine  consists  of  two 
horizontal  double  acting  cyl- 
inders coupled  to  the  ends  of 
the  same  axle  at  right  an- 
gles. These  are  located  un- 
der the  front  end  of  the 
boiler,  which  is  of  the  type 
described  in  Art.  563.  A 
pair  of  heavy  frames  sup- 
ports the  boiler,  the  load  be- 
ing carried  on  the  axles  by 
means  of  an  intervening 
"  spring  rigging."  The  stack 
is  necessarily  short,  so  that 
artificial  draft  is  provided  by 
means  of  an  expanding  noz- 
zle in  the  "smoke  box," 
through  which  the  exhaust 
steam  passes;  live  steam 
may  be  used  when  necessary 
to  supplement  this.  The 
engines  are  non-condensing, 
but  superheating  and  heat- 
ing of  feed  water,  particu- 
larly the  former,  are  being 
introduced  extensively.  The 
water  is  carried  in  an  aux- 
iliary tender,  excepting  in 
light  locomotives,  in  which  a 
"  saddle  "  tank  may  be  built 
over  the  boiler. 

The  ability  of  a  locomo- 
tive to  start  a  load  depends 
upon  the  force  which  it  can 
exert  at  the  rim  of  the  driv- 
ing wheel.  If  d  is  the  cylin- 
der diameter  in  inches,  L  the 
stroke  in  feet,  and  p  the 
maximum  mean  effective 
pressure  of  the  steam  per 
square  inch,  the  work  done 
per  revolution  by  two  equal 
cylinders  is  ird^Lp.  Assume 


THE  LOCOMOTIVE 


307 


this  work  to  be  trans- 
mitted to  the  point  of 
contact  between  wheel 
and  rail  without  loss, 
and  that  the  diameter 
of  the  wheel  is  D  feet, 
then  the  tractive  power, 
the  force  exerted  at 
the  rim  of  the  wheel, 


"KIT  ,H~ 


The  value  of  p,  with 
such  valve  gears  as  are 
employed    on  locomo- 
tives, may  be  taken  at 
80  to  85  per  cent  of  the 
boiler    pressure.      The 
actual  tractive  power, 
and  the    pull    on   the 
drawbar,   are    reduced 
by  the  friction  of  the 
mechanism  ;   the  latter 
from  5  to  15  per  cent. 
Under     ordinary    con- 
ditions    of     rail,     the 
wheels  will  slip  when 
the  tractive  power  ex- 
ceeds 0.22  to  0.25  the 
total  weight  carried  by 
the     driving      wheels. 
This    fraction    of    the 
total   weight  is   called 
the  adhesion,  and  it  is 
useless    to    make    the 
tractive  power  greater. 
In  locomotives  of  cer- 
tain types,  a  "  traction 
increaser  "  is  sometimes 
used.     This  is  a  device 
for  shifting  some  of  the 
weight  of  the  machine 
from  trailer  wheels  to 
driving   wheels.      The 
weight  on  the  drivers 
and   the   adhesion   are 
thereby  increased.  The 
engineman,    upon    ap- 


-""•"  --»,«- —prv- 

L'M 


308  APPLIED  THERMODYNAMICS 

preaching  a  heavy  grade,  may  utilize  a  higher  boiler  pressure  or  a  later  cut-off 
than  would  otherwise  be  useful. 

510.  Compounding.     Mallet  compounded  the  two  cylinders  as  early  as  1876. 
The  steam  pipe  between  the  cylinders  wound  through  the  smoke  box,  thus  becom- 
ing a  reheating  receiver.     Mallet  also  proposed  the  use  of  a  pair  of  tandem  compound 
cylinders  on  each  side.     The  Baldwin  type  of  compound  has  two  cylinders  on  each 
side,  the  high  pressure  being  above  the  low  pressure.     Webb  has  used  two  ordinary 
outside  cylinders  as  high-pressure  elements,  with  a  very  large  low-pressure  cylinder 
placed  under  the  boiler  between  the  wheels.     In  the  Cole  compound,  two  outside 
low-pressure  cylinders  receive  steam  from  two  high-pressure  inside  cylinders.     The 
former  are  connected  to  crank  pins,  as  in  ordinary  practice:  the  latter  drive  a 
forward  driving  axle,  involving  the  use  of  a  crank  axle.     The  four  crank  efforts 
differ  in  phase  by  90°.     This  causes  a  very  regular  rotative  impulse,  whence  the 
name  balanced  compound.     Inside  cylinders,  with  crank  axles,  are  almost  exclusively 
used,  even  with  simple  engines,  in  Europe :   two-cylinder  compounds  with  both 
cylinders  inside  have  been  employed.     The  use  of  the  crank  axle  has  been  complicated 
in  some  locomotives  with  a  splitting  of  the  connecting  rod  from  the  inside  cylinders 
to   cause  it  to  clear  the  forward  axle.     Greater  simplicity  follows  the  standard 
method  of  coupling  the  inside  cylinders  to  the  forward  axle. 

511.  Locomotive  Economy.     The  aim  in  locomotive  design  is  not  the  greatest 
economy  of  steam,  but  the  installation  of    the  greatest  possible  power-producing 
capacity  in  a  definitely  limited  space.     Notwithstanding  this,  locomotives  have 
shown  very  fair  efficiencies.     This  is  largely  due  to  the  small  excess  air  supply 
arising  from  the  high  rate  of  fuel  consumption  per  square  foot  of  grate  (Art.  564). 
The  locomotive's  normal  load  is  what  would  be  considered,  in  stationary  practice, 
an  extreme  overload.     Its  mechanical  efficiency  is  therefore  high.     For  the  most 
complete  data  on  locomotive  trials,  the  Pennsylvania  Railroad  Report  (47)  should 
be  consulted.     The  American  Society  of  Mechanical  Engineers  has  published  a 
code  (48)  ;  Reeve  has  worked  out  the  heat  interchange  in  a  specimen  test  by  Hirn's 
analysis  (49).     (See  Art.  554.) 

(1)  D.  K.  Clark,  Hallway  Machinery.  (2)  Isherwood,  Experimental  Researches 
in  Steam  Engineering,  1863.  (3)  De  la  condensation  de  la  vapeur,  etc.,  Ann.  des 
mines,  1877.  (4)  Bull,  de  la  Soc.  Indust.  de  Mulhouse,  1855,  et  seq.  (5)  Proc.  Inst. 
Civ.  Eng.,  CXXXII.  (6)  Peabody,  Thermodynamics,  1907,  233.  (7)  The  Engineer- 
ing Magazine,  December,  1906,  425.  (8)  Min.  Proc.  Inst.  C.  E.,  March,  1888  ;  April, 
1893.  (9)  Op.  cit.  (10)  Engine  Tests,  G.  H.  Barrus.  (11)  The  Steam  Engine, 
1892,  p.  190.  (12)  The  Steam  Engine,  1905,  109,  119,  120.  (13)  Proc.  Inst,  Mech. 
Eng.,  1889,  1892,  1895.  (14)  Ripper,  Steam  Engine  Theory  and  Practice,  1905,  p.  167. 
(15)  Ripper,  op.  cit.,  p.  149.  (16)  Trans.  A.  S.  M.  E.,  XXVIII,  10.  (17)  For  a 
discussion  of  the  interpretation  of  the  Boulvin  diagram,  see  Berry,  The  Temperature- 
Entropy  Diagram,  1905.  (18)  Proc.  Inst.  Mech.  Eng.,  January,  1895,  p.  132. 
(19)  The  Steam  Engine,  1906.  (21)  Trans.  A.  S.  M.  E.,  XV.  (22)  Ibid.,  XIII, 
647.  (23)  Ibid.,  XIX,  189.  (24)  Ibid.,  loc.  cit.  (25)  Ibid.,  XXV, 482, 483,  490,  492. 
(26)  Manuel  du  Conducteur  des  Machines  Binaires,  Lyons,  1850-1851.  (27)  Pea- 
body,  Thermodynamics,  1907,  283.  (28)  Thurston,  Engine  and  Boiler  Trials,  p.  130. 
(29)  Ripper,  Steam  Engine  Theory  and  Practice,  1905,  p.  412.  (30)  Experimental 


THE  STEAM   ENGINE  309 

Engineering,  1907.  (31)  The  Steam  Engine  Indicator,  1898.  Reference  should  also 
be  made  to  Miller's  and  Hall's  chapters  of  Practical  Instructions  for  using  the  Steam 
Engine  Indicator,  published  by  the  Crosby  Steam  Gage  and  Valve  Company,  1905. 
(32)  Low,  op.  cit.,  pp.  103-107  ;  Carpenter,  op.  cit.,  pp.  41-55,  531,  780.  (33)  Op.  cit., 
p.  391.  (34)  Trans.  A.  S.  M.  E.,  VI,  716.  (35)  Ibid.,  XXV.  (36)  76id.,'1892,  also 
XXV,  827.  (37)  Ibid.,  XI.  (38)  Ibid.,  XXIV,  713.  (39)  Op.  cit.,  144.  (40)  The 
Steam  Engine,  p.  212.  (41)  Bull,  de  la  Soc.  Ind.  de  Mulhouse,  1873.  (42)  Expose 
Succinct,  etc.  ;  Bevue  Universelle  des  Mines,  1880.  (43)  Carpenter,  Experimental 
Engineering,  1907,  657  ;  Peabody,  Thermodynamics,  1907,  225.  (44)  Trans.  A.  S. 
M.  E.,  XXVIII,  2,  225.  (45)  Zeuner,  Technical  Thermodynamics  (Klein),  II,  449. 
(46)  The  Steam  Engine,  1905,  I,  2,  3.  (47)  Locomotive  Tests  and  Exhibits  at  the 
Louisiana  Purchase  Exposition,  1906.  (48)  Trans.  A.  S.  M.  E.,  1892.  (49)  Ibid., 
XXVIII,  10,  1658. 

SYNOPSIS   OF   CHAPTER  XIII 
Practical  Modifications  of  the  Bankine  Cycle 

With  valves  moving  instantaneously  at  the  ends  of  the  stroke,  the  engine  would  operate 
in  the  non-expansive  cycle.  The  introduction  of  cut-off  makes  the  cycle  that  of 
Bankine,  modified  as  follows  :  — 

(1)  Port  friction  reduces  the  pressure  during  admission,  theoretically  along  a  line  of 
constant  total  heat.     This  dries  or  superheats  the  steam,  but  causes  a  loss  of  availa- 
bility of  the  heat.     The  piston  speed  influences  the  shape  of  the  admission  line. 
Regulation  by  throttling  is  wasteful. 

(2)  The  expansion  curve  differs  in  shape  and  position  from  that  in  the  ideal  cycle. 
Expansion  is  not  adiabatic.    The  steam  at  the  point  of  cut-off  contains  from  25 
to  70  per  cent  of  water  on  account  of  initial  condensation.    Further  condensation 
occurs  early  in  the  expansion   stroke,  followed  by  reevaporation  later  on,  after 
the  pressure  has  become  sufficiently  lowered.    The  inner  surfaces  only  of  the  walls 
fluctuate  in  temperature.     Condensation  is  influenced  by 

(a)  the  temperature  range  :  wide  limits,  theoretically  desirable,  introduce  some 

practical  losses  ; 

(6)  the  size  of  the  engine  :  the  exposed  surface  is  proportionately  greater  in 
•    small  engines  ; 

(c)  its  speed  :  high  speed  gives  less  time  for  heat  transfers  ; 

(d)  the  ratio  of  expansion  :   wide  ratios  increase   condensation   and  decrease 
efficiency,  particularly  because  of   increased  initial  condensation.     Initial 
wetness  facilitates  the  formation  of  further  moisture.     In  good  design,  the 
ratio  should  be   fixed  to  obtain  reasonably  complete   expansion   without 

excessive  condensation,  say  at  4  or  5  to  1.     M  =  —L-=.-\!s—. 


Steam  jackets  provide  steam  insulation  at  constant  temperature  ;  they  oppose  initial 
condensation  in  the  cylinder  and  are  generally  used  with  slow  speeds  and  high 
ratios  of  expansion.  Some  saving  is  always  shown.  Superheat,  used  under  similar 
conditions,  increases  the  mean  temperature  of  heat  absorption.  Each  75°  of  super- 
heat may  increase  the  dryness  at  cut-off  by  10  per  cent.  Superheat  increases 
efficiency,  and  is  preferable  to  increased  initial  pressure.  The  actual  expansion 


310  APPLIED.  THERMODYNAMICS 


curve,  PV  =  pv,  crosses  the  adiabatic.     M.  E.  P.  =  «      -  pd  with  the 

1T  ,,       2  x  diagram  factor  x  mALN 
Rankme  form  of  cycle.     H.  P.  =  ~33000  — 

(3)  The  exhaust  line  shows  lack  pressure  due  to  friction  of  ports,  the  presence  of  air, 
and  reevaporation.     High  altitudes  increase  the  capacity  of  non-condensing  engines. 

(4)  Clearance  varies  from  2   to   10  percent.     "Real  "and   "apparent"  ratios  of 
expansion. 

(5')  Compression  brings  the  piston  to  rest  quietly  ;  though  theoretically  less  desirable 
than  jacketing,  it  may  reduce  initial  condensation  if  properly  limited. 

(6)  Valve  action  is  not  instantaneous,  and  the  corners  of  the  diagram  are  always  some- 
what rounded. 

The  Steam  Engine  Cycle  on  the  Entropy  Diagram 

Cushion  steam,  present  throughout  the  cycle,  is  not   included  in  measurements   of 

steam  used. 
Its  volumes  may  be  deducted,  giving  a  diagram  representing  the  behavior  of  the 

cylinder  feed  alone. 
The   indicator  diagram   shows  actions  neither    cyclic   nor  reversible  :    it   depicts  a 

varying  mass  of  steam. 
The  Boulvin    diagram  gives  the  NT  history  correctly  along  the  expansion  curve 

only. 
The  Eeeve  diagram  eliminates  the  cushion  steam  ;   it  correctly  depicts  both   expan- 

sion and  compression  curves,  as  referred  to  the-  cylinder  feed. 
Diagrams  may  show  (a)  loss  by  condensation,  (6)  gains  by  increased  pressure  and 

decreased  back  pressure,  (c)  gains  by  superheating  and  jacketing. 

Multiple  Expansion 

Increased  initial  pressure  and  decreased  back  pressure  pay  best  with  wide  expansive 

ratios. 

Such  ratios  are  possible,  with  multiple  expansion,  without  excessive  condensation. 
Condensation   is   less  serious  because  of  (a)  the  use  made  of  reevaporated   steam, 

(&)  the   decrease   in   initial   condensation,  and  (c)  the   small  size  of  the  high- 

pressure  cylinder. 
Several  numbers  and  arrangements  of  cylinders  are  possible  with  expansion  in  two, 

three,  or  four  stages. 
Incidental  advantages  :  less  steam  lost  in  clearance  space  ;  compression  begins  later  ; 

the  large  cylinder  is  subjected  to  low  pressure  only  ;   more  uniform  speed  and 

moderate  strains  are  possible. 
The  Woolf  engine  had  no  receiver  ;  the  low-pressure  cylinder  received  steam  through- 

out the  stroke  as  discharged  by  the  high-pressure  cylinder.    The  former,  therefore, 

worked  without  expansion.     The  piston  phases  coincided  or  differed  by  180°. 
In  the  receiver  engine,  the  pistons  may  have  any  phase  relation  and  the  low-pressure 

cylinder  works  expansively.    Early  cut-off  in  the  low-pressure  cylinder  increases 

its  proportion  of  the  load,  and  is  practically  without  effect  on  the  total  work  of  the 

engine. 


THE  STEAM   ENGINE  311 

The  point  of  low-pressure  cut-off  to  eliminate  drop  may  be  graphically  or  analytically 
determined  for  tandem  and  cross-compound  engines. 

The  methods  given  ignore  angularity  of  the  connecting  rod,  clearance,  and  friction  in 
passages ;  they  assume  all  expansive  paths  to  be  hyperbolic. 

In  combining  diagrams,  two  saturation  curves  are  necessary,  unless  the  cushion  steam 
be  deducted. 

The  diagram  factor  has  an  approximate  value  the  same  as  that  in  a  simple  engine  hav- 
ing \/n  expansions,  in  which  n  is  the  number  of  expansions  in  the  compound 
engine  and  c  its  number  of  expansive  stages. 

Cylinder  ratios  are  3  or  4  to  1  if  non-condensing,  4  or  6  to  1  if  condensing,  in  com- 
pounds ;  triples  have  ratios  from  1 :  2.0 : 2.0  to  1 :  2.5 :  2.5.  A  large  high-pressure 
cylinder  gives  high  overload  capacity. 

The  engine  may  be  designed  by  computing  the  m.  e.  p.  of  the  combined  ideal  diagrams 
and  dividing  this  between  the  cylinders  so  as  to  equalize  work  areas,  or  by  assum- 
ing the  cylinder  ratio,  the  maximum  practicable  value  of  which  is  related  to  the 
total  ratio  of  expansion. 

Governing  should  be  by  varying  the  point  of  cut-off  in  both  cylinders. 

Drop  in  any  but  the  last  cylinder  is  usually  considered  undesirable. 

Exceptionally  high  efficiency  is  shown  by  compounds  having  cylinder  ratios  of  7  to  1. 
The  high-pressure  cylinder  in  ordinary  compounds  is  too  large  for  highest  efficiency. 

The  binary  vapor  engine  employs  the  waste  heat  of  the  exhaust  to  evaporate  a  fluid 
having  a  lower  boiling  point  than  can  be  attained  with  steam.  Additional  work 
may  then  be  evolved  down  to  a  rejection  temperature  of  60  or  70°  F.  The  best 
result  achieved  is  167  B.  t.  u.  per  Ihp.-minute. 

Engine  Tests 

The  indicator  measures  pressures  and  volumes  in  the  cylinder  and  thus  shows  the 

"cycle." 
Its  diagram  gives  the  m.  e.  p.  and  points  out  errors  in  valve  adjustment  or  control. 

Calorimeters :  the  barrel  type  :  #0  =  —          — ^j —  —  5 

wh\  +  Who  —  wh  —  Who 

surface  condensing :  XQ  =  —  —  — ; 

M  -Lo 

JJ    I      7./  "Y  £}  /j« 

superheating :  xQ  =  —        ^   r  —  '•>  limits  of  capacity  ; 

Barrus :  xn  = 


separating  :  direct  weighing  of  the  steam  and  water ; 
chemical :  insolubility  of  salts  in  dry  steam  ; 
electrical :  1  B.  t.  u.  =  17.59  watts  per  minute. 

Engine  trials  :  we  may  measure  either  the  heat  absorbed  or  the  heat  rejected  +  the  work 

done. 

By  measuring  both,  we  obtain  a  heat  balance. 
Results  usually  stated  :  Ib.  dry  or  actual  steam  per  Ihp.-hr.;  B.  t.  u.  per  Ihp.-minute  ; 

thermal  efficiency  ;   work  per  Ib.  ^.team ;   Carnot  efficiency  ;   Clausius  efficiency  ; 

efficiency  ratios. 


312  APPLIED  THERMODYNAMICS 

By  assuming  the  steam  dry  at  compression  and  release,  and  knowing  the  clearance,  we 
may  roughly  estimate  steam  consumption  from  the  indicator  diagram.  Reasonable 
accuracy  is  possible  if  the  quality  of  steam  at  these  points  be  known ;  no  informa- 
tion is  then  necessary  other  than  that  given  by  the  diagrams  themselves. 

Duty  =  ft.-lb.  of  work  per  100  Ib.  coal.    Plant  efficiency  =  B>  t>  u'  of  work  .     Mechani- 
*  ,  B.  t.  u.  in  coal 

cal  efficiency  =*™*e^    . 
Indicated  hp. 

Hirri*s  analysis:  Ex  =  2Jf (hx  +  xxrx};  HX  —  EX  +  Wx;  heat  transfer  to  and  from 
walls  may  be  computed  from  the  supply  of  heat,  the  change  in  internal  energy, 
and  the  work  done.  The  excess  of  losses  over  gains  represents  radiation. 


Types  of  Steam  Engine 


The  pulsometcr  :  efficiency  =  s(y  -f  1)  -r-  (xoLo  -f  /?o  —  AI). 

Standard  engines  :  non-condensing  or  condensing  ;  right-hand  or  left-hand  ;  simple 
or  multiple  expansion  ;  single-acting  or  double-acting  ;  rotative  or  non-rotative  ; 
duplex  or  single  ;  horizontal,  vertical,  or  inclined  ;  locomotive,  stationary  (pump- 
ing, mill,  power  plant),  or  marine  ;  belted,  direct-connected,  or  rope-drive  ;  air 
compressors  ;  girder,  tangye  or  semi-tangye  frames  ;  slow,  medium,  or  high  speed  ; 
throttling,  automatic,  four-valve,  or  releasing  gear. 

The  power  plant  :  feed  pump,  boiler,  engine,  condenser. 


The  locomotive  :  tractive  power  =          ;  adhesion  =  0.22  to  0.25  x  weight  on  drivers  ; 

two-cylinder  and  four-cylinder  compounds  ;  the  balanced  compound  ;  high  econ- 
omy of  locomotive  engines. 


PROBLEMS 

1.  Show  from  Art.  426  that  the  loss  by  a  throttling  process  is  equal  to  the  prod- 
uct of  the  increase  of  entropy  by  the  absolute  temperature  at  the  end  of  the  process. 

2.  Ignoring  radiation,  how  fast  are  the  walls  gaining  heat  because  of  transfers 
during  expansion  in  an  engine  running  at  100  r.  p.  m.,  in  which  ^  pound  of  steam  is 
condensed  per  revolution  at  a  mean  pressure  of  100  Ib.,  and  0.30  pound  is  reevaporated 
at  a  mean  pressure  of  42  Ib  ? 

3.  Establish   from  Art.  434  an  approximate   formula  for  the   relation  between 
engine  speed  and  wetness  at  cut-off  in  one  of  the  tests. 

4.  All  other  factors  being  the  same,  how  much  less  initial  condensation,  at  \  cut- 
off, should  be  found  in  an  engine  30|"  x  48"  than  in  one  7"  x  7"  ? 

5.  Sketch  a  curve  showing  the  variation  in  engine  efficiency  with  ratio  of  expan- 
sion. 

6.  Find  the  percentage  of  initial  condensation  at  £  cut-off  in  an  engine  using  dry 
steam,  running  at  100  r.  p.  m.  with  a  pressure  at  cut-off  of  120  Ib. ,  the  engine  being 
30|"  x  48"  (Art.  437). 

7.  In  Fig.  193,  assuming  the  initial  pressure  to  have  been  100  Ib.,  the  feed-water 
temperature  90°  F.,  find  the  approximate  thermal  efficiencies  with  the  various  amounts 
of  superheat  at  a  load  of  15  hp. 


THE  STEAM   ENGINE  313 

8.  In  an  ideal  Clausius  cycle  with  initially  dry  steam  between  p  =  140  and  p  =  2 
(Art.  417),  by  what  percentage  would  the  efficiency  be  increased  if  the  initial  pressure 
were  made  160  Ib.  ?     By  what  percentage  would  it  be  decreased  if  the  lower  pressure 
were  made  6  Ib.  ? 

9.  Find  the  mean  effective  pressure  in  the  ideal  cycle  with  hyperbolic  expansion 
and  no  clearance  between  pressure  limits  of  120  and  2  Ib.,  with  a  ratio  of  expansion  of  4. 

10.  Find  the  probable  indicated  horse  power  of  a  double-acting  engine  with  the 
best  type  of  valve  gear,  jackets,  etc.,  operating  as  in  Problem  9,  at  100  r.  p.  ni.,  the 
cylinder  being  30£"  x  48".     (Ignore  the  piston  rod.) 

11.  In  Problem  9,  what  percentage  of  power  is  lost  if  the  lower  pressure  is  raised 
to  3|  Ib.  ? 

12.  By  what  percentage  would  the  capacity  of  an  engine  be  increased  at  an  altitude 
of  10,000  ft.  as  compared  with  sea  level,  at  120  Ib.  initial  pressure  and  a  back  pressure 
1  Ib.  greater  than  that  of  the  atmosphere,  the  ratio  of  expansion  being  4  ?     (Atmos- 
pheric pressure  decreases  £  Ib.  per  1000  ft.  of  height.) 

13.  An  engine  has  an  apparent  ratio  of  expansion  of  4,  and  a  clearance  amounting 
to  0.05  of  the  piston  displacement.     What  is  its  real  ratio  of  expansion  ? 

14.  In  the  dry  steam  Clausius  cycle  of  Problem  8,  by  what  percentages  are  the  ca- 
pacity and  efficiency  affected  if  expansion  is  hyperbolic  instead  of  adiabatic  ?    Discuss 
the  results. 

15.  In  the  dry  steam  cycle  of  Problem  9,  find  the  change  in  capacity  and  efficiency 
if  the  cycle  is  worked  with  hyperbolic  compression  to  one  fourth  the  initial  pressure, 
clearance  equal  to  5  per  cent  of  the  piston  displacement,  hyperbolic  expansion,  1  Ib.  of 
mean  wiredrawing  during  admission,  70  per  cent  decrease  in  volume  at  cut-off  due  to 
initial  condensation,  and  2  Ib.  of  mean  extra  back  pressure  during  exhaust. 

16.  In  an  engine  having  a  clearance  volume  of  1.0  and  a  back  pressure  of  2  Ib., 
the  pressure  at  the  end  of  compression  is  40  Ib.    If  the  compression  curve  is  PF1-03  =  c, 
what  is  the  volume  at  the  beginning  of  compression  ? 

17.  An  engine  works  between  120  and  2  Ib.  pressure,  the  piston  displacement 
being  20  cu.  ft.,  clearance  5  per  cent,  and  apparent  ratio  of  expansion  4.     The  expan- 
sion curve  is  PF1-0'2  =  c,  the  compression  curve  PF1'03  =  c,  and  the  final  compression 
pressure  is  40  Ib.     Plot  the  PF  diagram  with  actual  volumes  of  the  cushion  steam 
eliminated. 

18.  In  Problem  16,  1.825  Ib.  of  steam  are  present  per  cycle.     Plot  the  entropy  dia- 
gram from  the  indicator  card  by  Boulvin's  method.     Plot  the  F.Y  diagram. 

19.  In  Problems  17  and  18,  compute  and  plot  the  entropy  diagram  by  Reeve's 
method,  assuming  the  steam  dry  at  the  beginning  of  compression.     (See  Art.  394.) 
Discuss  any  differences  between  this  diagram  and  that  obtained  in  Problem  18. 

20.  In  a  non-expansive  cycle  with  dry  steam  at  cut-off  and  no  clearance,  find  the 
changes  in  capacity  and  economy  by  raising  the  initial  pressure  from  100  to  120  Ib., 
the  back  pressure  being  2  Ib. 

21.  A  non-expansive  engine  with  limiting  volumes  of  1  and  6  cu.  ft.  and  an  initial 
pressure  of  120  Ib.,  without  compression,  has  its  back  pressure  decreased  from  4  to  2  Ib. 
Find  the  changes  in  capacity  and  efficiency.    The  same  steam  is  now  allowed  to  expand 
hyperbolically  to  a  volume  of  21  cu.  ft.     Find  the  effects  following  the  reduction  of 
back  pressure  in  this  case.    The  steam  is  in  each  case  dry  at  the  point  of  cut-off. 


314  APPLIED  THERMODYNAMICS 

22.  Find  the  cylinder  dimensions  of  an  automatic  engine  to  develop  30  horse 
power  at  300  r.  p.  m.,  non-condensing,  at  ^  cut-off,  the  initial  pressure  being  100  Ib. 
and  the  piston  speed  300  ft.  per  minute.    The  engine  is  double-acting. 

23.  Sketch  a  possible  cylinder  arrangement  for  a  quadruple-expansion  engine  with 
seven  cylinders,  three  of  which  are  vertical  and  four  horizontal,  showing  the  receivers 
and  pipe  connections. 

24.  Using  the  ideal  combined  diagram  for  a  compound  engine  with  a  constant 
receiver  pressure,  clearance  being  ignored,  what  must  that  receiver  pressure  be  to 
divide  the  diagram  area  equally,  the  pressure  limits  being  120  and  2  and  the  ratio  of 
expansion  16  ? 

25.  Consider  a  simple   engine   30|"  x  48"   and  a  compound   engine   15£"   and 
30£"  x  48",  all  cylinders  having  5  per  cent  of  clearance  and  no  compression.     What 
are  the  amounts  of  steam  theoretically  wasted  in  filling  clearance  spaces  in  the  simple 
engine  and  in  the  high-pressure  -cylinder  of  the  compound,  the  pressures  being  as  in 
Problem  24  ? 

26.  Take  the  same  engines.     The  simple  engine  has  a  real  ratio  of  expansion  of  4  ; 
the  compound  is  -as  in  Problems  24  and  25.     Compression  is  to  be  carried  to  40  Ib.  in 
the  simple  engine  and  to  60  Ib.  in  the  compound  in  order  to  prevent  waste  of  steam. 
By  what  percentages  are  the  work  areas  reduced  in  the  two  engines  under  consideration  ? 

27.  A  cross-compound  double-acting  engine  operates  between  pressure  limits  of 
120  and  2  Ib.  at  100  r.  p.  m.  and  800  ft.  piston  speed,  developing  1000  hp.     Find  the 
sizes  of  the  cylinders  under  the  following  assumptions,  there  being  no  drop  :    (a)  dia- 
gram factor  0.85,   20  expansions,  receiver  pressure  24  Ib.  ;  (6)  diagram  factor  0.85, 
20  expansions,  work  equally  divided  ;  (c)  diagram  factor  0.85,  20  expansions,  cylinder 
ratio  5:1;  (d)  diagram  factor  0.83,  32  expansions,  work  equally  divided.     Find  the 
power  developed  by  each  cylinder  in  (a)  and  (c).     Find  the  size  of  the  cylinder  of  the 
equivalent  simple  engine  having  a  diagram  factor  of  0.85  with  20  expansions.    Draw  up 
a  tabular  statement  of  the  five  designs  and  discuss  their  comparative  merits. 

28.  In  Problem  27,  Case  (a),  the  receiver  volume  being  equal  to  that  of  the  high- 
pressure  cylinder,  find  graphically  and  analytically  the  point  of  cut-off  on  the  low- 
pressure  cylinder. 

29  a.  Find  the  point  of  cut-off,  as  in  Problem  28,  if  the  engine  is  a  tandem  com- 
pound with  5  Ib.  of  drop. 

29  &.    In  what  respects  are  the  results  in  Problems  27,  28,  and  29  a  to  be  modified 
so  as  to  include  the  factors  in  Art.  473  and  Art.  474  ? 

30.  Trace  the  combined  diagram  for  one  end  of  the  cylinder  from  the  first  set  of 
cards  in  Fig.  230,  assuming  the  clearance  in  each  cylinder  to  have  been  15  per  cent  of 
the  piston  displacement,  the  cylinder  ratio  3  to  1,  and  the  pressure  scales  of  both  cards 
to  be  the  same. 

31.  In  Fig.  204  assume  the  steam  to  have  been  70  per  cent  dry  at  cut-off,  95  per 
cent  dry  at  the  beginning  of  compression  in  the  high-pressure  cylinder,  and  90  per 
cent  dry  at  the  beginning  of  compression  in  the  low-pressure  cylinder,  the  cylinder 
ratio  being  4  ;  and  plot  the  combined  diagram  with  cushion  steam  eliminated,  showing 
the  single  saturation  curve. 

32.  Show  on  the  entropy  diagram  the  effect  of  reheating. 


THE  STEAM   ENGINE  315 

33.  In  Art.  483,  what  was  the  Carnot  efficiency  of  the  Josse  engine  ?     Assuming 
it  to  have  been  used  in  combination  with  a  gas  engine,  the  maximum  temperature  in 
the  latter  being  3000°  F.,  by  what  approximate  amount  might  the  Carnot  efficiency 
have  been  increased  ?     (The  temperature  of  saturated  sulphur  dioxide  at  35  Ib.  pres- 
sure is  52°  F.) 

34.  An  indicator  diagram  has  an  area  of  82,192.5  foot-pounds.     What  is  the  mean 
effective  pressure  if  the  engine  is  30|"  x  48"  ?     What  is  the  horse  power  of  this  engine 
if  it  runs  double-acting  at  100  r.  p.  m  ? 

35.  Given  points  1,  2  on  a  hyperbolic  curve,  such  that  V-2—Vi  =  15,  PI  =  120, 
Po  =  34.3,  find  the  OP-axis. 

36.  An  engine  develops  500  hp.  at  full  load,  and  62  hp.  when  merely  rotating  its 
wheel  without  external  load.     What  is  its  mechanical  efficiency  ? 

37.  Steam  at  100  Ib.  pressure  is  mixed  with  water  at  100°.     The  weight  of  water 
increases  from  10  to  11  Ib.,  and  its  temperature  rises  to  107 1°.     What  was  the  percent- 
age of  dryness  of  the  steam  ? 

38.  The  same  steam  is  condensed  in  and  discharged  from  a  coil,  its  temperature 
becoming  210°,  and  10  Ib.  of  surrounding  water  rise  in  temperature  from  100°  to  204£°. 
Find  the  quality  of  the  steam.     What  would  have  been  an  easier  way  of  determining 
the  quality  ? 

39.  What  is  the  maximum  percentage  of  wetness  that  can  be  measured  in  a  throt- 
tling calorimeter  in  steam  at  100  Ib.  pressure,  if  the  discharge  pressure  is  30  Ib.  ? 

40.  Steam  at  100  Ib.  pressure  has  added  to  it  from  an  external  source  30  B.  t.  u. 
per  pound.     It  is  throttled  to  30  Ib.  pressure,  its  temperature  becoming  270.3°.     What 
was  its  dryness  ? 

41.  In  Problem  40,  the  added  heat  is  from  an  electric  current  of  5  amperes  pro- 
vided for  one  minute,  the  voltage  falling  from  220  to  110.     What  was  the  amount  of 
heat  added  and  the  percentage  of  dryness  of  the  steam  ? 

42.  An  engine  consumes  10,000  Ib.  of  dry  steam  per  hour,  the  moisture  having 
been  completely  eliminated  by  a  receiver  separator  which  at  the  end  of  one  hour  is 
found  to  contain  285  Ib.  of  water.     What  was  the  dryness  of  the  steam  entering  the 
separator  ? 

43.  Check  all  results  that  can  be  checked  in  Arts.  498,  499,  500. 

A  double-acting  engine  at  100  r.p.  m.  and  a  piston  speed  of  800  feet  per  minute 
gives  an  indicator  diagram  in  which  the  pressure  limits  are  120  and  2  Ib.,  the  volume 
limits  1  and  21  cu  ft.  The  apparent  ratio  of  expansion  is  4.  The  expansion  curve 
follows  the  law  PF1-02  =  c.  Compression  is  to  40  Ib.,  according  to  the  law  PF1-03  =  c. 
Disregard  rounded  corners.  The  boiler  pressure  is  130  Ib.,  the  steam  leaving  the  boiler  is 
dry,  the  steam  at  the  throttle  being  95  per  cent  dry  and  at  120  Ib.  pressure.  The  boiler 
evaporates  26,500  Ib.  of  steam  per  hour ;  2000  Ib.  of  steam  are  supplied  to  the  jackets 
at  120  Ib.  pressure.  The  engine  runs  jet-condensing,  the  inlet  water  weighing  530,000 
Ib.  per  hour  at  43.85°  F.,  the  outlet  weighing  554,000  Ib.  at  90°  F.  The  coal  burned  is 
2700  Ib.  per  hour,  its  average  heating  value  being  14,000  B.  t.  u.  Compute  as  follows : 

44  a.  The  mean  effective  pressure  and  indicated  horse  power.  (NOTE.  The  work 
quantities  under  the  curves  must  be  computed  with  much  accuracy.) 

44  b.    The  cylinder  dimensions  of  the  engine. 


316  APPLIED  THERMODYNAMICS 

45.  The  heat  supplied  at  the  throttle  per  pound  of  cylinder  and  jacket  steam,  and 
the  B.  t.  u.  consumed  per  Ihp.  per  minute  ;  the  engine  being  charged  with  heat  above 
the  temperature  of  the  condenser  discharge  (Art.  502). 

46.  The  dry  steam  consumption  per  Ihp.-hr.,  thermal  efficiency,  and  work  per 
pound  of  dry  steam. 

47.  The  Carnot  efficiency,  the  Clausius  efficiency,  and  the  efficiency  ratio,  taking 
the  limiting  conditions  as  at  the  throttle  and  the  condenser  outlet. 

48.  The  cylinder  feed  steam  consumption  computed  as  in  Art.  500  ;  the  consump- 
tion thus  computed  but  assuming  x  =  0.80  at  release,  x  =  1.00  at  compression.     Com- 
pare with  Problem  46. 

49.  The  steam  consumption  computed  as  in  Art.  501 ;  develop  the  expression 

3,960,000  (  WE  -  VT)  (1  +  a) 

144  (P  +  p) 

for  indicated  steam  consumption  in  a  simple  engine  giving  the  same  diagram  at  both 
ends  of  the  cylinder  and  having  the  same  clearance  at  each  end. 

50.  The  percentage  of  steam  lost  by  leakage  (all  leakage  occurring  between  the 
boiler  and  the  engine)  ;  the  transmissive  efficiency  ;  the  unaccounted-for  losses. 

51.  The  duty,  the  efficiency  of  the  plant,  and  the  boiler  efficiency. 

52.  The  heat  transfers  and  the  loss  of  heat  by  radiation,  as  in  Art.  504,  assuming 
x  =  1.00  at  compression.     Compare  the  unaccounted-for  heat  with  that  obtained  in 
Problem  50. 

53.  The  value  of  the  mechanical  equivalent  of  heat  which  might  be  computed  from 
the  experiment. 

54.  A  pulsoineter  receives  water  at  its  own  level  and  lifts  it  30  feet.     The  dis- 
charge being  at  a  temperature  of  190°,  and  0.004  Ib.  of  dry  steam  being  supplied  per 
pound  of  water  lifted,  at  100  Ib.  pressure,  find  the  efficiency. 

55.  Explain  the  meaning  of  the  figure  2068.84  in  Art.  503. 

56.  Revise  Fig.  233,  showing  the  arrangement  of  machinery  and  piping  if  a  surface 
condenser  is  used. 

57.  A  locomotive  weighing  200,000  Ib.  carries,  normally,  60  per  cent  of  its  weight 
on  its  drivers.     The  cylinders  are  19"  x  26",  the  wheels  66"  in  diameter.     What  is 
the  maximum  boiler  pressure  that  can  be  profitably  utilized  ?    If  the  engine  has  a  trac- 
tion increaser  that  may  put  12,000  Ib.  additional  weight  on  the  drivers,  what  maximum 
boiler  pressure  may  then  be  utilized  ? 

58.  What  is  the  percentage  of  error  in  the  calculation  of  Art.  500  ? 

59.  Represent  Fig.  217  on  the  PV  diagram. 


CHAPTER   XIV 


FIG.  235. 


Arts.  512,  524,  536.  — De  Laval  Turbine 
Wheel  and  Nozzles. 


THE   STEAM  TURBINE 

512.  The  Turbine  Principle.     Figure  235  shows  the  method  of  using  steam  in 
a  typical  impulse  turbine.     The  expanding  nozzles  discharge  a  jet  of  steam  at  high 
velocity  and  low  pressure  against 

the   blades   or    buckets,   the  ini- 

pulse  of    the    steam   causing  ro-  s\  fv-.1 

tation.       We     have     here,     not  Jpf\\«l  \ 

expansion  of  high  pressure  steam 
against  a  piston,  as  in  the  ordi- 
nary engine,  but  utilization  of 
the  kinetic  energy  of  a  rapidly 
flowing  stream  to  produce  move- 
ment. One  of  the  assumptions 
of  Art.  11  can  now  no  longer 
hold.  All  of  the  expansion  oc- 
curs in  the  nozzle  ;  the  expansion 
produces  velocity,  the  velocity  does 
work.  The  lower  the  pressure 
at  which  the  steam  leaves  the  nozzle,  the  greater  is  the  velocity  attained.  It  will 
presently  be  shown  that  to  fully  utilize  the  energy  of  velocity,  the  buckets  must 
themselves  move  at  a  speed  proportionate  to  that  of  the  steam.  This  involves  ex- 
tremely high  rotative  speeds. 

The  steps  in  the  design  of  an  impulse  turbine  are  (a)  determination 
of  the  velocity  produced  by  expansion,  (6)  computation  of  the  nozzle 
dimensions  necessary  to  give  the  desired  expansion,  and  (c)  the  propor- 
tioning of  the  buckets. 

513.  Expansive  Path.     There  is  a  gradual  fall  of  pressure  while  the 
steam  passes  through  the  nozzle.     With  a  given  initial  pressure,  the  pres- 
sure and  temperature  at  any  stated  point  along  the  nozzle  should  never 
change.     There  is,  therefore,  no  tendency  toward  a  transfer  of  heat  be- 
tween steam  and  walls.     Further,  the  extreme  rapidity  of  the  movement 
gives  no  time  for  such  transfer ;  so  that  the  process  in  the  nozzle  is  truly 
adiabatic,  although  friction  renders  it  non-isentropic.     The  first  problem 
of  turbine  design  is  then  to  determine  the  changes  of  velocity,  volume, 
temperature  or  dryness,  and  pressure,  during  such  adiabatic  expansion, 
for  a  vapor  initially  wet,  dry,  or  superheated ;  the  method  may  be  accu- 

317 


318  APPLIED  THERMODYNAMICS 

rate,  approximate  (exponential),  or  graphical.     The  results  obtained  are 
to  include  the  effect  of  nozzle  friction. 

514.  The  Turbine  Cycle.     Taking  expansion  in  the  turbine  as  adiabatic 
and  as  carried  down  to  the  condenser  pressure,  the  cycle  is  that  of  Clausius, 
and  is  theoretically  more  efficient  than  that  of  any  ordinary  steam  engine 
working  through  the  same  range.     The  turbine  is  free  from  losses  due  to 
interchange  of  heat  with  the  ivalls.     The  practical  losses  are  four  : 

(a)  Friction  in  the  nozzles,  causing  a  fall  of  temperature  without  the 
performance  of  work  ; 

(6)  Incomplete  utilization  of  the  kinetic  energy  by  reason  of  the 
assumed  blade  angles  and  residual  velocity  of  the  emerging  jet  (Art.  528); 

(c)  Friction  along  the  buckets,  increasing  as  some  power  of  the  stream 
speed  ; 

(d)  Mechanical  friction  of  journals  and  gearing,  and  friction  between 
steam  and  rotor  as  a  whole. 

515.  Heat  Loss  and  Velocity.     In  Fig.  236,  let  a  fluid  flow  adiabatically 
from  the  vessel  a  through  the  frictionless  orifice  b.     Let  the  internal  en- 

ergy of  the  substance  be  e  in  a  and  E  in  b  ;  the 
velocities  v  and  V\  the  pressures  p  and  P;  and 
the  specific  volumes  w  and  W.  If  the  velocities 
could  be  ignored,  as  in  previous  computations, 
the  volume  of  each  pound  of  fluid  in  a  would 
decrease  by  w  in  passing  out  at  the  constant 
pressure  j?;  and  the  volume  of  each  pound  of 

FIG.  2:16.  Art.  515.  —  Flow  fl^  jn  &  would  increase  by  W  at  the  constant 
pressure  P.  The  net  external  work  done  would 

be  PW—pw,  the  net  loss  of  internal  energy  e  —  E,  and  these  two  quan- 

tities would  be  equal.     With  appreciable  velocity  effects,  we  must  also 

consider  the  kinetic  energies  in  a  and  b  ;  these  are 

tf        ,   F2 
—  and  £-1 

2y  2g 

and  we  now  have 

H=T+  I+W+V, 

(T+T)+W+V=0, 


or  -L  =piv  -  PW+e  -  E. 

2g      2g 


HEAT  DROP  AND   VELOCITY 


319 


Let  X,  U,  H,  R,  and  x,  u,  A,  r,  be  the  dry  ness,  increase  of  vol- 
ume during  vaporization,  heat  of  liquid,  and  internal  latent  heat,  at 
PW&ndpw  respectively  ;  let  s  be  the  specific  volume  of  water;  then 
for  expansion  of  a  vapor  from  pw  to  P  W  within  the  saturated  region, 


- 


in  which  q,  Q  represent  total  heats  of  wet  vapor  above  32  degrees. 
If  expansion  proceeds  from  the  superheated  to  the  saturated  region, 


in  which  n  =  u  +  s  is  the  volume  of  saturated  steam  at  the  pressure  p, 
iv  is  the  volume  of  superheated  steam,  and 

p(w  -  n) 


WD 


is  the  internal  energy  measured  above  saturation.*     This  also   re- 

duces to  q  —  Q  -f-  s(p  —  P),  where  q  is  the  total  heat  in  the  super- 

heated  steam,  and  the  same  form  of 

expression  will  be  found  to  apply  to 

expansion  wholly  in  the  superheated 

region.      The  gain  in  kinetic  energy 

of  a  jet  due  to  adiabatic  expansion  to 

a  lower  pressure  is  thus  equivalent  to 

the  decrease  in  the  total  heat  of  the 

steam  plus  the  work  which  would  be 

required    to    force    the    liquid    back 

against  the   same   pressure  head.     In 

Fig.   237,  let  ab,   AB,    CD,  represent  the  three  paths.     Then  the 

losses  of  heat  are  represented  by  the  areas  dabc,  deABc,  deCDfc. 

*  For  any  gas  treated  as  perfect,  the  gain  of  internal  energy  from  t  to  T  is 


FlG-  237- 


Art.  sw.-  Adiabatic  Heat 


i(r-t)=*(r-g)=^g-(r-o  =      ~{ 

or  in  this  case,  since  internal  energy  is  gained  at  constant  pressure, 


320  APPLIED  THERMODYNAMICS 

The  term  s(p  —  P)  being  ordinarily  negligible,  these  areas  also  rep- 
resent the  kinetic  energy  acquired,  which  may  be  written, 

Z!_^=?_(?. 

"ig      2g     q 

In  the  turbine  nozzle,  the  initial  velocity  may  also,  without  serious 
error,  be  regarded  as  negligible  ;  whence 


,  _ 
2~  =  q-  #or  V=  V50103.2  (gr-  #)  =  223.84  Vj-  §  feet  per  second. 

516.  Computation  of  Heat  Drop.    The  value  of  q  —  Q  may  be  determined 
for  an  adiabatic  path  between  stated  limits  from  the  entropy  diagram, 
Fig.  175,  or  from  the  Mollier  diagram,  Fig.  177.     Thus,  from  the  last 
named,  steam  at  100  Ib.  absolute  pressure  and  at  500°  F.  contains  1273 
B.  t.  u.  per  pound;    steam  85  per  cent   dry  at  3  Ib.  absolute   pressure 
contains  973  B.  t.  u.     Steam  at  150  Ib.  absolute  pressure  and  600°  F.  con- 
tains 1317  B.  t.  u.     If  it  expand  adiabatically  to  2.5  Ib.  absolute  pressure, 
its  condition  becomes  88  per  cent  dry,  its  heat  contents  1000  B.  t.  u.,  and 
the  velocity  produced  is 

223.84  V317  =  4000  ft.  per  second. 

517.  Vacuum  and  Superheat.     The  entropy  diagram  indicates  the  nota- 
ble gain  due  to  high  vacua  and  superheat.     Comparing  dry  steam  expanded 
from  150  Ib.  to  4  Ib.  absolute  pressure  with  the  same  steam  superheated 
to  600°  and  expanded  to  2.5  Ibs.  absolute  pressure,  we  find  q  —  Q  in  the 
former  case  to  be  63  B.  t.  u.,  and  in  the  latter,  317  B.  t.  u.     The  corre- 
sponding values  of  "Fare  1770  and  4000  ft.  per  second.     The  turbine  is 
peculiarly  adapted  to  realize  the  advantages  of  wide  ratios  of  expansion. 
These  do  not  lead  to  an  abnormally  large  cylinder,  as  in  ordinary  engines  ; 
the  "  toe  "  of  the  Clausius  diagram,  Fig.  184,  is  gained  by  allowing  the 
steam  to  leave  the  nozzle  at  the  condenser  pressure.     Superheat,  also,  is 
not  utilized  merely  in  overcoming  cylinder  condensation  ;  it  increases  the 
available  "  fall  "  of  heat,  practically  without  diminution. 

518.  Effect  of  Friction.    If  the  steam  emerging  from  the  nozzle  were  brought 
back  to  rest  in  a  closed  chamber,  the  kinetic  energy  would  be  reconverted  into 
heat,  as  in  a  wiredrawing  process,  and  the  expanded  steam  would  become  super- 
heated.    Watkinson  has,  in  fact,  suggested  this  (1)  as  a  method  of  superheating 
steam,  the  water  being  mechanically  removed  at  the  end  of  expansion,  before  re- 
conversion to  heat  began.     In  the  nozzle,  in  practice,  the  friction  of  the  steam 
against  the  walls  does  partially  convert  the  velocity  energy  back  to  heat,  and  the 
heat  drop  and  velocity  are  both  less  than  in  the  ideal  case. 


EFFECT   OF   NOZZLE  FRICTION 


321 


In   Fig.  238,  for  adiabatic  expansion  from  p,  v,  q,  to  P,   V,  Qf  the 
velocity  imparted  is 


223.84  -Vq  -  Q. 

During  expansion  from  p,  v,  q,  to  P,,  F1?  ft, 
the  velocity  imparted  is 


223.84  -Vq-Qt. 

Since  Fi  exceeds  V,  the  steam  is  more  nearly 
dry  at  Pi;  i.e.  Ql  exceeds  Q.  The  loss  of 
energy  due  to  the  path  pvq  —  PiViQi  as 
compared  with  peg  —  PVQ,  is 


P.V.Q, 


FIG.  238.  Art.  518.  —  Abiabatic 
Expansion  with  and  without 
Friction. 


FIG.    239.       Art.    518.  —  Expansive 
Path  as  Modified  by  Friction. 


in  which  X2  is  the  difference  of  the  squares  of  the  velocities  at  Q  and  ft. 

This  gives  X2  =  50103.2  (ft  -  Q).  In  Fig.  239,  let  NA  be  the  adiabatic 

path,  NX  the  modified  path  due  to  fric- 
tion. NZ  represents  a  curve  of  constant 
total  heat ;  along  this,  no  work  would  be 
done,  but  the  heat  would  steadily  lose  its 
availability.  As  NX  recedes  from  NA 
toward  NZ,  the  work  done  during  expan- 
sion decreases.  Along  NA,  all  of  the  heat 
lost  (area  FHNA)  is  transformed  into 
work;  along  NZ,  no  heat  is  lost  and  no 
work  is  done,  the  areas  BFHNC  and 
BFZD  being  equal.  Along  NX,  the  heat 

transformed    into   work   is   BFHNC  -  BFXE  =  FHNA  -  CAXE,  less 

than  that  during  adiabatic  expansion  by  the  amount  of  work  converted 

back  to  heat.     Considering  expansion  from  N  to  Z. 

V=  223.84 

since  q  =  ft.     Nozzle   friction   decreases  the  heat   drop,   the   final   velocity 
attained,  and  the  external  work  done. 

519.  Allowance  for  Friction  Loss.  For  the  present,  we  will  assume 
nozzle  friction  to  reduce  the  heat  drop  by  10  per  cent.  In  Fig.  240,  which 
is  an  enlarged  view  of  a  portion  of  Fig.  177,  let  AB  represent  adiabatic 
(isentropic)  expansion  from  the  condition  A  to  the  state  B.  Lay  off 


AB 


322 


APPLIED  THERMODYNAMICS 


and  draw  the  line  of  constant  heat  CD. 


FIG.  240. 


and 


Arts.  519,  524,  525,  532,  534.  —  The  Steam  Path 
of  the  Turbine. 


Then  D  is  the  equivalent  final 
state  at  the  same  pressure 
as  that  existing  at  B,  and 
AC    represents    the    heat 
drop  corrected  for  friction. 
Similarly  by  laying  off 
HQ==AH 
=  10 

and  drawing  GE  to  inter- 
sect the  35-lb.  pressure 
line,,  we  find  the  point  E 
on  the  path  AD  of  the 
steam  through  the  nozzle. 
We  may  use  the  new  heat 
drop  thus  obtained  in  de- 
termining F;  or  generally, 
if  m  is  the  friction  loss, 


V=  223.84  VI  -  m  Vq  -  Q. 


If  m  =  0.10,  V  =  212.42  V?  -  Q. 

520.  Analytical  Relations.  The  influence  of  friction  in  determining  the  final 
condition  of  the  steam  may  be  examined  analytically.  For  example,  let  the  initial 
condition  be  wet  or  dry;  then  friction  will  not  ordinarily  cause  superheating,  so 
that  the  steam  will  remain  saturated  throughout  expansion.  Without  friction,  the 
final  dryness  XQ  would  be  given  by  the  equation  (Art.  392), 

,        t      xl     xJn 

l°e-T+-t=T?- 

Friction  causes  a  return  to  the  steam  of  the  quantity  of  heat  m(q  —  Q).     This  in- 


creases the  final  dryness  by 


,  making  it 


<o 

If  the  initial  condition  is  superheated  to  ts,  and  the  final  condition  saturated, 
adiabatic  expansion  would  give 

t      I      ,,       tt      xnl0 


+  m(q  - 


and  friction  would  make  the  final  condition 

r/ioo-  '  + l 
x°  = I 


NOZZLE   PROPORTIONS 


323 


If  the  steam  is  superheated  throughout  expansion,  we  have  for  the  final  tem- 
perature Ts,  without  friction, 

t  I  f,  1Q  Ts 

J.  t  t  J_  1 

in  which  the  value  of  k0  must  be  obtained  by  successive  approximations. 

521.   Rate  of  Flow.     For  a  flow  of  G  pounds  per  second  at  the  velocity  F,when 

GW 

the  specific  volume  is  W,  the  necessary  cross-sectional  area  of  nozzle  is  F  = 

The  values  of  W  and  V  may  be 
read  or  inferred  from  the  heat 
chart  or  the  formulas  just  given. 
In  Fig.  241  (2),  let  ab  represent 
frictionless  adiabatic  expansion 
on  the  TN  plane,  a'b'  the  same 
process  on  the  PV  plane.  By 
finding  qa  and  values  of  Q  at 
various  points  along  ab,  we  may 
obtain  a  series  of  successive 
values  of  V.  The  correspond- 
ing values  of  W  being  read  from 
a  chart  or  computed,  we  plot  the 
curve  MN,  representing  the  re- 
lation of  specific  volume  and 
velocity  throughout  the  expan- 
sion. Draw  yy'  parallel  to  OW, 
making  Oy  =  G,  to  some  con- 
venient scale.  Draw  any  line  OD  from  O  to  MN,  intersecting 


FIG.  241. 


Art.  521.  — Graphical  Determination  of 
Nozzle  Area. 


at  k.     From 


similar  triangles,  yk :  yO :  :  On  :  nD,  or  yk  = 


GW 


To  find  the  pressure  at  any  specified  point  on  the  nozzle,  lay  off  yk  —  F,  draw 
OkD,  Dn,  and  project  z  to  the  PT  plane.  The  minimum  value  of  F  is  reached 
when  OD  is  tangent  to  MN.  It  becomes  infinite  when  V  =  0.  The  conclusion 
that  the  cross-sectional  area  of  the  nozzle  reaches  a  minimum  at  a  certain  stage  in  the 
expansion  will  be  presently  verified. 


522.   Maximum  Flow.     For  a  perfect  gas, 


PW 


If  the  initial  velocity  be  negligible,  we  have,  as  the  equation  of  flow  (Art.  515), 


y-i 


and  since 


y-i 


324  APPLIED  THERMODYNAMICS 

Then 


From  Art.  521, 


Taking  the  value  of  V  at 


we  obtain 


G  = 


m 


,pv 
IP) 


i- 

J 


This  reaches  a  maximum,  for  air,  when  P  -4-  p  =  0.5274  (3).  The  velocity  is  then 
equal  to  that  of  sound.  For  dry  steam,  on  the  assumption  that  y  —  1.135,  and 
that  the  above  relations  apply,  the  ratio  for  maximum  flow  is  0.577. 

Using  the  value  just  given  for  the  ratio  P  -H-  p,  with  y  =  1.402,  the  equation 
for  G  simplifies  to 


G  =  0.491  -, 

*  Rt 

the  equation  of  flow  of  a  permanent  gas,  which  has  been  closely  confirmed  by 
experiment.  With  steam,  the  ratio  of  the  specific  heats  is  more  variable,  and  the 
ratio  of  pressures  has  not  been  as  well  confirmed  experimentally.  Close  approxi- 
mations have  been  made.  Clarke  (4),  for  example,  shows  maximum  flow  with 
saturated  steam  to  occur  at  an  average  ratio  of  0.56.  The  pressure  of  maximum 
flow  determines  the  minimum  or  throat  diameter  of  the  nozzle,  which  is  independ- 
ent of  the  discharge  pressure.  The  emerging  velocity  may  be  greater  than  that 
in  the  throat  if  the  steam  is  allowed  to  further  expand  after  passing  the  throat. 
The  nozzle  should  in  all  cases  continue  beyond  the  throat,  either  straight  or  ex- 
panding, if  the  kinetic  energy  is  all  to  be  utilized  in  the  direction  of  flow. 

523.  Experiments.  Many  experiments  have  been  made  on  the  flow  of  fluids 
through  nozzles  and  orifices.  Those  of  Jones  and  Rathbone  (5),  Rosenhain  (6), 
Gutermuth  (7),  Napier  (8),  Rateau  (9),  Hall  (10),  Wilson  (11),  Kunhardt  (12), 
Buchner  (13),  Kneass  (14),  Lewicki  (15),  Durley  (16),  and  chiefly,  perhaps,  those 
of  Stodola  (17),  should  be  studied.  There  is  room  for  further  advance  in  our 
knowledge  of  the  friction  losses  in  nozzles  of  various  proportions.  There  are  sev- 
eral methods  of  experimentation:  the  steam,  after  passing  the  orifice,  may  be  con- 
densed and  weighed;  the  pressure  at  various  points  in  the  nozzle  may  be  measured 
by  side  orifices  or  by  a  searching  tube  ;  or  the  reaction  or  the  impulse  of  the  steam 
at  its  escape  may  be  measured.  The  velocity  cannot  be  measured  directly. 


TYPES   OF  TURBINE 


325 


A  greater  rate  of  flow  is  obtainable  through  an  orifice  in  a  thin  plate  (Fig. 
242)  than  through  an  expanding  nozzle  (Fig.  243).     For  pressures  under  80  lb., 
with  discharge  into  the  atmosphere,  the  plain  orifice  is  more  efficient 
in  producing  velocity.      For  wider  pressure  ranges,  a  divergent 
nozzle   is   necessary   to   avoid   deferred   expansion  occurring   after 
emergence.      Expansion  should  not,  however,  be  carried  to  a  pres- 
sure loM*er  than  that  of  discharge.     The  rate  of  flow,  but  not  the 
emerging  velocity,  depends  upon  the  shape  of  the  inlet  ;    a  slightly 
rounded  edge  (Fig.  243)  gives  the  greatest  rate  ;  a  greater  amount    FIG.  242.    Art. 
of  rounding  may  be  less  desirable.     The  experimentally  observed     523.—  Diverg- 

critical   pressure  ratio  (—  ,  Art.  522)  ranges  with   various   fluids 
\p  / 

from  0.50  to  0.85.     Maximum  flow  occurs  at  the  lower  ratios  with  rather  sharp 
corners  at  the  entrance,  and  at  the  higher  ratios  when  a  long  divergence  occurs 
beyond  the  throat,  as  in  Fig.  243.     The  "  most  efficient  " 
nozzle  will  have  different  proportions  for  different  pressure 
ranges.     The  pressure  is,  in  general,  greater  at  all  points 
along  the  nozzle  than  theory  would  indicate,  on  account  of 
FIG.    243.     Arts.    523,  friction  ;  the  excess  is  at  first  slight,  but  increases  more  and 
525.  —  Expanding  more  rapidly  during  the  passage.     Most  experiments  have 
necessarily  been  made  on  very  small  orifices,  discharging  to 

the  atmosphere.  The  friction  losses  in  larger  orifices  are  probably  less.  The 
experimental  method  should  include  at  least  two  of  the  measurements  above 
mentioned,  these  checking  each  other.  The  theory  of  the  action  in  the  nozzle 
has  been  presented  by  Heck  (18).  Zeuner  (19)  has  discussed  the  flow  of  gases  to 
and  from  the  atmosphere  (20),  both  under  adiabatic  and  actual  conditions,  and 
the  efflux  of  gases  in  general  through  orifices  and  long  pipes. 


BUCKET  WHEEl        I 

8 


524.  Types  of  Turbine.  The  single  stage  impulse  turbine  of  Fig. 
235  is  that  of  De  Laval.  Its  action  is  illustrated  in  Fig.  244.  The 
pressure  falls  in  the  nozzle,  and  remains 

PRESSURES 

constant  in  the  buckets.     The  Curtis  and 
..........     Rateau    turbines 

use     a     series    of 

wheels,    with    ex- 

panding   nozzles 

between    the    va-   FlG 

rious  series  (Figs. 

245»  246).     The  steam  is  only  partially  ex- 
FIG.  245.     Art.  524.  —  Curtis  panded  in  each  nozzle,  until  it  reaches  the 

Turbine.  „      ,  ,   . 

last  one.     Such  turbines  are  of  the  multi- 

stage impulse  type.      During  passage  through  the  blades,  the  ve- 
locity decreases,  while   the   pressure    remains   unchanged.     In  the 


PATH   OF 
STEAM 


Axt-  524.  -De  Laval 

Turbine. 


326 


APPLIED  THERMODYNAMICS 


pressure  turbine  of  Parsons,  there  are  no  expanding  nozzles  ;  the 
steam  passes  successively  through  the  stationary  guide  vanes  (r,  #, 
ENSURES , .  and  movable  wheel  buckets,  TT,  w,  Fig.  247. 
^  A  gradual  fall  of  pressure  occurs,  the  buck- 
1*^1!! OF  ets  being  at  all  times  full  of  steam.  In 
impulse  turbines,  the  buckets  need  not  be 
full  of  steam,  and  the  pressure  drop  occurs 

FIG.  246.      Art.  524.-Rateau    in  the  nozzle  Only. 

A  lower  rotative  speed  results  from  the 

use  of  several  pressure  stages  with  expanding  nozzles.  Let  the 
total  heat  drop  of  317  B.  t.  u.,  in  Art. 
516,  be  divided  into  three  stages  by  three 
sets  of  nozzle's.  The  exit  velocity  from 
each  nozzle,  corrected  for  friction,  is 
then  212.42V?-  Q  =  2180  ft.  per  sec- 
ond, instead  of  3790  ft.  per  second ;  lay-  FIG.  247.  Arts.  524,  533.  — Parsons 
ing  off  in  Fig.  240  the  three  equal  heat 

drops,  we  find  that  the  nozzles  expand  between  150  and  50,  50  and 
13,  arid  13  and  2.5  Ib.  respectively.  The  rotative  speeds  of  the 
wheels  (proportional  to  the  emerging  velocities),  Art.  528,  are  thus 
reduced. 

525.  Nozzle  Proportions ;  Volumes.  The  specific  volume  W  of  the 
steam  at  any  point  along  the  path  AD,  Fig.  240,  having  been  obtained 
from  inspection  of  the  entropy  chart,  or  from  the  equation  of  condition, 
and  the  velocity  V  at  the  same  point  having  been  computed  from  the 

WG 

heat  drop,  the  cross-sectional  area  of  the  nozzle,  in  square  feet,  is  F=  • 

(Art.  521).  Finding  values  of  F  for  various  points  along  the  expansive 
path,  we  may  plot  the  nozzle  as  in  Fig.  243,  making  the  horizontal  inter- 
vals, ab,  be,  cd,  etc.,  such  that  the  angle  between  the  diverging  sides  is 
about  10°,  following  standard  practice.  It,  has  been  shown  that  F  reaches 
a  minimum  value  when  the  pressure  is  about  0.57  of  the  initial  pres- 
sure, and  then  increases  as  the  pressure  falls  further.  If  the  lowest 
pressure  exceeds  0.57  of  the  initial  pressure,  the  nozzle  converges  toward 
the  outlet.  Otherwise,  the  nozzle  converges  and  afterwards  expands,  as 
in  Fig.  243.  Let,  in  such  case,  o  be  the  minimum  diameter,  0  the  outlet 
diameter,  L  the  length  between  these  diameters;  then  for  an  angle  of 

10°  between  the  sides,  Q  - 1  =  L  tan  5°,  or  L  =  5.715(0  -  o). 


VELOCITY  DIAGRAMS 


327 


526.  Work  Done.     The  work  done  in  the  ideal  cycle  per  pound 
of  steain  is  778((?  —  Q)  foot-pounds.     Since  1  horse  power  =  1,980,000 
foot-pounds  per  hour,  the  steam  consumption  per  hp.-hr.  is  theoreti- 
cally  1,980,000  -f-  778(2  -<?)  =  2545  -*-(?-<?)•     If   E   is   the  effi~ 
ciency   ratio  of    the   turbine,    from    steam    to   buckets,    and   e   the 
efficiency  from  steam  to  shaft,  then  the  actual  steam  consumption 
per  indicated  horse  power  is  2545  -*-  E(q  —  (>),  and  per  brake  horse 
power  is  2545  -r-  e(q  —  Q)  pounds.    The  modifying  influences  of  nozzle 
and  bucket  friction  in  determining  E  are  still  to  be  considered. 

527.  Relative  Velocities.     In  Fig.  248,  let  a  jet  of  steam  strike 
the  bucket  A  at  the  velocity  t>,  the  bucket  itself  moving  at  the  speed 
u.     The  velocity  of  the  steam  rela- 
tive  to   the    bucket   is   then   repre- 
sented in  magnitude  and  direction 

by  V.  The  angles  a  and  e  made 
with  the  plane  of  rotation  of  the 
bucket  wheel  are  called  the  absolute 
entering  and  relative  entering  angles 
respectively.  Analytically,  sin  e  =  v 
sin  a-s-V.  The  stream  traverses 


FIG.  248.    Art.  527.  —  Velocity  Diagram. 


the  surface  of  the  bucket,  leaving  it  with  the  relative  velocity  a/, 
which  for  convenience  is  drawn  as  x  from  the  point  0.  Without 

bucket  friction,  x  =  V.  The 
angle  /  is  the  relative  angle  of 
exit.  Laying  off  w,  from  z,  we 
find  Y  as  the  absolute  exit  ve- 
locity, with  g  as  the  absolute 
angle  of  exit.  Then,  if  x  =  F, 
sin  g  =  V  sin  /  -r-  Y. 

To  include  the  effect  of  nozzle 
and  bucket  friction,  we  proceed 

FIG.  249.    Arts.  527,  532,  534.  —  Velocity       as  in  Fig.   249,  decreasing  v  to 
Corrected  for  Friction.  Vl  -  m    of    its    original    value 

(Art.  519),  and  making  x  less  than  Fby  from  5  to  20  per  cent,  as 
in  ordinary  practice.  As  before,  sin  e  =  v  sin  a  -j-  F;  but  for  a  bucket 
friction  of  10  per  cent,  sin  g  —  0.9  V  sin/ -^  Y. 


328 


APPLIED  THERMODYNAMICS 


528.  Bucket  Angles  and  Work  Done.  In  Fig.  250,  the  absolute 
velocities  v  and  Y  may  be  resolved  into  components  ab  and  db  in  the 
direction  of  rotation,  and  ac  and  de  at  right 
angles  to  this  direction.  The  former  compo- 
nents are  those  which  move  the  wheel ;  the  lat- 
ter produce  an  end  thrust  on  the  shaft.  Now 
ab  -f  bd  (bd  being  negative)  is  the  change  in 
velocity  of  the  fluid  in  the  direction  of  rotation ; 
it  is  the  acceleration ;  the  force  exerted  per 
pound  is  then 

(ab  +  bd)  +g=(ab  +  bd)  -*-  32.2 

=  (v  cos  a  4-  Y  cos  g)  -r-  32.2. 

This  force  is  exerted  through  the  distance  u 
the  work  done  per  pound  of  steam  is  then 
•*-  32.2  foot-pounds.  This,  from  Art.  526,  equals 


FIG.  250.  Arts.  528, 529.  — 
Rotative  and  Thrust 
Components. 


feet  per  second  ; 
u(v  cos  a  +  l^cos  g 
778  E  (q  -  Q)  whence 

E=u(v  cos  a  +  Tcos^)  -H  25051. 6(q  -  Q). 
The  efficiency  is  thus  directly  related  to  the  bucket  angles. 

To  avoid  splashing,  the  entrance  angle  of  the  bucket  is  usually 
made  equal  to  the  relative  entering  angle  of  the  jet,  as  in  Fig.  251. 
(These  formulas  hold  only  when  the  sides  of  the 
buckets  are  enclosed  to  prevent  the  lateral 
spreading  of  the  stream.)  In  actual  turbines, 
bd  (Fig.  250)  is  often  not  negative,  on  account 
of  the  extreme  reversal  of  direction  that  would 
be  necessary.  With  positive  values  of  bd,  the 
maximum  work  is  obtained  as  its  value  ap- 
proaches zero,  and  ultimately  it  is  uv  cos  a -r-32. 2. 

Since  the  kinetic  energy  of  the  jet  is   — ,  the    FIG.  251.    Art.  528.— 

2  g  Velocities  and  Bucket 

efficiency  E  from  steam  to  buckets  then  becomes       Angles- 

In  designing,  we  may  either  select  an  exit  bucket  angle 


fit 

2  -  cos  a. 


which  shall  make  bd  equal  to  zero  (the  relative  exit  velocity  being 
tangential  to  the  surface  of  the  bucket),  or  we  may  choose  such  an 
angle  that  the  end  thrust  components  de  and  ca,  Fig.  250,  shall  bal- 


VELOCITY   EFFICIENCY 


329 


ance.  In  marine  service,  some  end  thrust  is  advantageous ;  in 
stationary  work,  an  effort  is  made  to  eliminate  it.  This  would  be 
accomplished  by  making  the  entrance  and  exit  bucket  angles  equal, 
for  a  zero  retardation  by  friction.  With  friction  considered,  the 
angle  of  exit  K,  in  Fig.  251,  must  be  greater  than  the  entering  an- 
gle e.  In  any  case,  where  end  thrust  is  to  be  eliminated,  the  rota- 
tive component  of  the  absolute  exit  velocity  must  be  so  adjusted  as 
to  have  a  detrimental  effect  on  the  economy. 

529.  Effect  of  Stream  Direction  on  Efficiency.  Let  the  stream  strike 
the  bucket  in  the  direction  of  rotation,  so  that  the  angle  «  =0,  Fig.  250, 
the  relative  exit  velocity  being  perpendicular 
to  the  plane  of  the  wheel.  The  work  done  is  -« — 


V  —  U 


9 


,  while  the  kinetic  energy  is  - — -|     The 


rt2      •  o 

emciency,   2  u 


v  —  u 


becomes   a   maximum   at 


0.50  when  u  =  -  •     With  a  cup-shaped  vane,  as 

in  the  Pelton  wheel,  Fig.  252,  complete  reversal  ^ 

of  the  jet  occurs  ;   the  absolute  exit  velocity, 

ignoring   friction,  is  v-2u.     The  change  in  FIG.  252.    Arts.  529,  536.-  Pel- 

,      .  ,      .  N          ,  ,  ,  ton  Bucket. 

velocity  is  v  +  v  —  2u  =  2(v  —  u),  and  the  work 


is  2u(v—  n)  -s-  g,  whence  the  efficiency, 


becomes  a  maximum 


at  100  per  cent  when  u =  -•     Complete  reversal  in  turbine  buckets  is  im- 
practicable. 

530.    Single-Stage  Impulse  Turbine.     The  absolute  velocity  of  steam  enter- 
ing the  buckets  is  computed  from  the  heat  drop  and  nozzle  friction  losses.     In  a 

turbine  of  this  type,  the  speed  of  the 
buckets  can  scarcely  be  made  equal 
to  half  that  of  the  steam ;  a  more 
usual  proportion  is  0.3.  The  velocity 
u  thus  seldom  exceeds  1400  ft.  per 
second.  Fixing  the  bucket  speed  and 
the  absolute  entering  angle  of  the 
steam  (usually  20°)  we  determine 
graphically  the  entering  angle  of  the 
bucket.  The  bucket  may  now  be  de- 
signed with  equal  angles,  which  would 
eliminate  end  thrust  if  there  were  no 
FIG.  253.  Art.  530.  — Bucket  Outline.  friction,  or,  allowance  being  made  for 


330 


APPLIED  THERMODYNAMICS 


friction,  either  end  thrust  or  the  rotative  component  of  the  absolute  exit  velocity 
may  be  eliminated.  The  normals  to  the  tangents  at  the  edges  of  the  buckets  being 
j  drawn,  as  ec,  Fig.  253, 

the  radius  r  is  made 
equal  to  about  0.965  ec. 
The  thickness  t  may 
be  made  equal  to  0.2 
times  the  width  kl. 
The  bucket  as  thus 
drawn  is  to  a  scale  as 
yet  undetermined; 
the  widths  kl  vary  in 
practice  from  0.2  to 
1.0  inch. 

It  should  be  noted 
that  the  back,  rather 
than  the  front,  of  the 
bucket  is  made  tan- 
gent to  the  relative 
velocity  V.  The  work 
per  pound  of  steam 
being  computed  from 
the  velocity  diagram, 
and  the  steam  con- 
sumption estimated 
for  the  assumed  out- 
put, we  are  now  in  a 
position  to  design  the 
nozzle. 


531.  Multi-stage 
Impulse  Turbine.  If 
the  number  of  pres- 
sure stages  is  few,  as 
in  the  Curtis  type,  the 
heat  drop  may  be  di- 
vided equally  between 
the  stages.  In  the 
Rateau  type,  with  a 
large  number  of 
stages,  a  proportion- 
ately greater  heat  drop 
occurs  in  the  low-pres- 
sure stages.  The  cor- 
responding intermedi- 


FlG.  254.     Art.  531.  —  Curtis  Turbine.     (General  Electric  Company.) 


ate  pressures  are  determined  from  the  heat  diagram,  and  the  various  stages  are 
then  designed  as  separate  single-stage  impulse  turbines,  all  having  the  same  rota- 


DESIGN   OF  MULTI-STAGE  TURBINE 


331 


tive  speed.  The  entrance  angles  of  the  fixed  intermediate  blades  in  the  Curtis 
turbine  are  equal  to  those  of  the  absolute  exit  velocities  of  the  steam.  Their  exit 
angles  may  be  adjusted  as  desired;  they  may  be  equal  to  the  entrance  angles  if 
the  latter  are  not  too  acute.  The  greater  the  number  of  pressure  stages,  the 
lower  is  the  economical  limit  of  circumferential  speed;  and  if  the  number  of 
revolutions  is  fixed,  the  smaller  will  be  the  wheel.  Figure  254  shows  a  recent 
form  of  Curtis  turbine,  with  five  pressure  stages,  each  containing  two  rows  of 
moving  buckets.  The  electric  generator  is  at  the  top. 

532.  Problem.  Preliminary  Calculations  for  a  Multi-stage  Impulse  Turbine. 
To  design  a  1000  (brake)  hp.  impulse  turbine  with  three  pressure  stages,  having 
two  moving  wheels  in  each  pressure  stage.  Initial  pressure,  150  Ib.  absolute; 
temperature,  600°  F. ;  final  pressure,  2  Ib.  absolute;  entering  stream  angles,  20°; 
peripheral  velocity,  500  ft.  per  second  ;  1200  revolutions  per  minute. 

By  reproducing  as  in  Fig.  240  a  portion  of  the  Mollier  heat  chart,  we  obtain 
the  expansive  path  AB,  and  the  heat  drop  is  1316.6  -  987.5  =  329.1  B.  t.  u.  Divid- 
ing this  into  three  equal  parts,  the  heat  drop  per  stage  becomes  329.1  -f-  3  =  109.7 
B.  t.  u.  This  is  without  correction  for  friction,  and  we  may  expect  a  somewhat 
unequal  division  to  appear  as  friction  is  considered.  To  include  friction  in  deter- 
mining the  change  of  condition  during  flow  through  the  nozzle,  we  lay  off,  in  Fig. 

240,  AH  =  109.7,  HG  =  — ,  and  project  GE,  finding p  =  50,  t  =  380°,  at  the  out- 
lets of  the  first  set  of  nozzles.  The  velocity  attained  (with  10  per  cent  loss  of 
available  heat  by  friction)  is  v  =  212.42  V109.7  =  2225  ft.  per  second. 


^--^  i  u         n  f 

FIG.  255.    Art.  532.  —  Multi-stage  Velocity  Diagram. 

We   now   lay  off  the  velocity  diagram,  Fig.  249,  making  a  =  20°,  u  =  500, 
v  =  2225.     The  exit  velocity  x  may  be  variously  drawn ;  we  will  assume  it  so  that 


332  APPLIED  THERMODYNAMICS 

the  relative  angles  e  and/  are  equal,  and,  allowing  10  per  cent  for  bucket  friction, 
will  make  x  =  0.9  F.  For  the  second  wheel,  the  angle  a'  is  again  20°,  while  v',  on 
account  of  friction  along  the  stationary  or  guide  blades,  is  0.9  Y.  After  locating 
F',  if  the  angles  e'  and/'  were  made  equal,  there  would  in  some  cases  be  a  back- 
ward impulse  upon  the  wheel,  tending  to  stop  it,  at  the  emergence  of  the  jet  along 
F.  On  the  other  hand,  if  the  angle/'  were  made  too  acute,  the  stream  would  be 
unable  to  get  away  from  the  moving  buckets.  With  the  particular  angles  and 
velocities  chosen,  some  backward  impulse  is  inevitable.  We  will  limit  it  by  mak- 
ing/' =  30°.  The  rotative  components  of  the  absolute  velocities  may  be  computed 
as  follows,  the  values  being  checked  as  noted  from  the  complete  graphical  solution 
of  Fig.  255  : 
ab  =  v  cos  20°  =  2225  x  0.93969  =  2090.81.  (2080) 

cd  =  cz  -  dz  =  0.9  Fcos/-  u  =  0.9  Fcose  -  u  =  0.9(2090.81  -  500)  -  500  =  931.73. 

(925) 
ef=  egcos2Q°  =  0.9  eg*  cos  20°  =  0.9  x  1158  x  0.93969  =  979.     (975) 

kl  =  km  -lm  =  500  -  x'  cos  30°  =  500  -  0.9  V  cos  30° 

=  500  -(0.9  x  596.2  f  x  0.86603)=  36. 

/  nb  +  cd  +  ef-kl\        3966  x  500      ft1  _nn 
The  work  per  pound  of  steam  is  then  (  -    —        •'      —\u—    —  —  —  — 

footpounds,  in  the  first  stage.  This  is  equivalent  to  61,500  +-  778  -  79.2  B.  t.  u. 
The  heat  drop  assumed  for  this  stage  was  109.7  B.  t.  u.  The  heat  not  converted 
into  work  exists  as  residual  velocity  or  has  been  expended  in  overcoming  nozzle 
and  bucket  friction  and  thus  indirectly  in  superheating  the  steam.  It  amounts 
to  109.7  -  79.2  =  30.5  B.  t.  u. 

Returning  to  the  construction  of  Fig.  240,  we  lay  off  in  Fig.  256  an  =  79.2 
B.  t.  u.  and  project  no  to  ko,  finding  the  condition  of  the  steam  after  passing  the 
first  stage  buckets.  Bucket  friction  has  moved  the  state  point  from  m  to  0,  at 
which  latter  point  Q  =  .1237.2,  p  =  50,  t  =  414°.  This  is  the  condition  of  the  steam 
which  is  to  enter  the  second  set  of  nozzles.  These  nozzles  are  to  expand  the  steam 
down  to  that  pressure  at  which  the  ideal  (adiabatic)  heat  drop  from  the  initial 
condition  is  2  x  109.7  =  219.4  B.  t.  u.  Lay  off  ae  =  219.4,  and  find  the  line  eg  of 
12  Ib.  absolute  pressure.  Drawing  the  adiabatic  op  to  intersect  eg,  we  find  the 
heat  drop  for  the  second  stage,  without  friction,  to  be  1237.2  —  1120  =  117.2  B.  t.  u., 
giving  a  velocity  of  212.42  Vll7^2  =  2299.66  ft.  per  second. 

*  To  find  cgr,  we  have 

cb  =  Fcos  e  =  2090.81  -  500  =  1590.81,  bj  =  v  sin  a  =  2225  x  0.34202  =  760.99, 
V=^d?+b?  =  Vi590.812  -i-  760.992  =  1765,  x  =  0.9  F=  0.9  x  1765  =  1588.5, 


ch  =  x  sin/  =  1588.5  sin  e  =  1588.  5^  =  1588.5  ™^?  =  685, 

F  17bo 


eg  =     ch   +  h     =  veS     +  oaT       =  1158. 

t  To  find  F',  we  have 

gf=  v'  sin  20°  =  0.9  Y  sin  20°  =  0.9  x  1158  x  0.34202  =  355, 

nf  =  ef  -u  =  979  -  500  =  479,  V  =  V^/*  +  jff*  =  V^  +  355*  =  590.2. 


STEAM   PATH,   MULTI-STAGE  TURBINE 


333 


j>=150 


The  complete  velocity  diagram  must  now  be  drawn  for  the  second  stage,  fol- 
lowing the  method  of  Fig.  255.  This  gives  for  the  rotative  components,  ab  =  2160.97, 
cd  =  994.87,  ef=  1032.59,  kl  =  8.06.  (There  is  no  backward  impulse  from  Id  in 
this  case.)  The  work  per  pound  of  steam  is 

500(2160.97  +  994.87  + 1032.59  +  8.06)  =  65  163 
32.2 

or  83.76  B.  t.  u.  Of  the  available  heat  drop,  117.2  B.  t.  u.,  33.44  have  been  ex- 
pended in  friction,  etc.  Laying  off,  in  Fig.  256,  pq  =  33.44,  and  projecting  qr  to 
meet  pr,  we  have  r  as  the  state  point  for  steam 
entering  the  third  set  of  nozzles.  Here  p  =  12, 
*!  =  223°,  Q\  =  1153.44.  In  expanding  to  the  ' 
final  condenser  pressure,  the  ideal  path  is  rs, 
terminating  at  2  Ib.  absolute,  and  giving  an  un- 
corrected  heat  drop  of  Qr-Qs  =  1153.44  -  1039 
=  114.44  B.  t.  u.  The  velocity  attained  is 
212.42  VlHM  =  2271.83  ft.  per  second.  A  third 
velocity  diagram  shows  the  work  per  pound  of 
steam  for  this  stage  to  be  63,823  foot-pounds,  or 
82.04  B.  t.  u.  We  are  not  at  present  concerned 
with  determining  the  condition  of  the  steam  at 
its  exit  from  the  third  stage. 

The  whole  work  obtained  from  a  pound  of 
steam  passing  through  the  three  stages  is  then 
79.2  +  83.76  +  82.04  =  245.0  B.  t.  u.  The  horse 
power  required  is  1000  at  the  brake  or  say 
1000  •*•  0.8  =  1250  hp.  at  the  buckets.  This  is 

equivalent  to  1250  x  198^000  =  3,181,250  B.  t.  u. 

7  t  8 

per  hour.  The  pounds  of  steam  necessary  per 
hour  are  3,181,250  •*•  245.0  =  12,974.  This  is 
equivalent  to  10.38  Ib.  per  brake  hp.-hr.,  a  result 
sufficiently  well  confirmed  by  the  test  results 
given  in  Chapter  XV. 

Proceeding    now  to   the   nozzle  design,  we 

adopt  the  formula  F  =  — —  from  Art.  521.     It 

FIG.  256.      Art.  532.  —  Steam  Path, 

will    be   sufficiently   accurate  to    compute  cross-  Multi-stage  Turbine, 

sectional  areas  at  throats  and  outlets  only.     The 

path  of  the  steam,  in  Fig.  256,  is  as  follows :  through  the  first  set  of  nozzles,  along 
am;  through  the  corresponding  buckets,  along  mo;  thence  alternately  through 
nozzles  and  buckets  along  ou,  ur,  rv,  vt.  The  points  u,  v,  etc.,  are  found  as  in  Fig. 
240.  It  is  not  necessary  to  plot  accurately  the  whole  of  the  paths  am,  ou,  rv;  but 
the  condition  of  the  steam  must  be  determined,  for  each  nozzle,  at  that  point  at 
which  the  pressure  is  0.57  the  initial  pressure  (Art.  522).  The  three  initial  pres- 
sures are  150,  50,  and  12;  the  corresponding  throat,  pressures  are  85.5,  28.5,  and 
6.84.  Drawing  these  lines  of  pressure,  we  lay  off,  for  example,  wx  =  ^  aw,  project 


334 


APPLIED   THERMODYNAMICS 


xy  to  wy,  and  thus  determine  the  state  y  at  the  throats  of  the  first  set  of  nozzles.    The 
corresponding  states  are  similarly  determined  for  the  other  nozzles.    We  thus  find, 

at  y,  p  =  85.5,  t  =  474°,  at  TO,  p  =  50,  t  =  380°, 
q  =  1260.5  ;  q  =  1217.87 ; 

&tA,p  =  28.5,  t  =  313°,  at  w,  p  =  12,  x  =  0.989, 
7=1192;  9  =  1131.72; 

at  B,  p  =  6.84,  x  =  0.9835,  at  v,  p  =  2,  x  =  0.932, 
9  =  1118;  9  =  1050.44. 

We  now  tabulate  the  corresponding  velocities  and  specific  volumes,  as  below. 
The  former  are  obtained  by  taking  F  =  223.84  V^  -  q2 ;  the  latter  are  computed  from 

the  Tumlirz  formula,  W  =  0.5963  —  -  0.256.     Thus,  at  the  throat  of  the  first  nozzle, 

V  =  223.84  V1316.6  -  1260.5  =  1683 ;  while  W  =  0.5963  46°  +  474  _  Q.256  =  6.26. 

8o.5 

In  the   wet  region,  the  Tumlirz  formula  is   used   to   obtain  the  volume  of  dry 
steam  at  the  stated  pressure  and  the  tabular  corresponding  temperature ;  this  is 
applied  to  the  wet  vapor :     Ww  =  0.017  +  x(  W0  -  0.017).     The  tabulation  follows. 
At  y,  V  =  1683,  W  =  6.26 ;  at  TO,  F  =  2225,  W  =  9.724  : 

at  A    V  =  1507,  W  =  15.92 ;  at  ti,  F  =  2299,  W  =  32.24 ; 

at  B,  V  =  1330,  W  =  53.92 ;  at  t;,   F  =  2271,  W  =  162.62. 

The  value  of  G,  the  weight  of  steam  flowing  per  second,  is  12,974  -=-  3600  =  3.604  Ib. 
For  reasonable  proportions,  we  will  assume  the  number  of  nozzles  to  be  16  in  the 
first  stage,  42  in  the  second,  and  180  in  the  third.  The  values  of  G  per  nozzle  for 
the  successive  stages  are  then  3.604  -s-  16  =  0.22525,  3.604  -s-  42  =  0.08581  and 
3.604  +  180  =  0.02002.  We  find  values  of  F  as  follows  : 


At 


at  TO, 


at  A, 


1683 
0.22525  x  9.724 

2225 
0.08581  x  15.92 


0.08581  x  82.24 


2299 


=  0.000989;    at  B, 


=  0.000903;    at  i», 


53'92 


1330 
0.02002x162.62 


=  0.000809  ; 


=  0.00144. 


1507  2271 

Completing  the  computation  as  to  the  last  set  of  nozzles  only,   the  throat 
area  is  0.000809  sq.  ft.,   that  at  the  outlet  being  0.00144  sq.  ft.     These  corre- 

spond to  diameters  of  0.385  and 
0.515  in.  The  taper  may  be  uniform 
from  throat  to  outlet,  the  sides  mak-* 
ing  an  angle  of  10°.  This  requires 
a  length  from  throat  to  outlet  of 
(0.515  -  0.385)  -*-  2  tan  5°  =  0.742  in. 
The  length  from  inlet  to  throat  may 
be  one  fourth  this,  or  0.186  in.,  the 
edge  of  the  inlet  being  rounded. 
The  nozzle  is  shown  in  Fig.  257. 


FIG.  257.    Art.  532.  —  Third  Stage  Nozzle. 


The  diameter  of  the  bucket  wheels  at  mid-height  is  obtained  from  the  rotative 
speed  and  peripheral  velocity.    Tf  d  be  the  diameter, 

3.1416  d  x  1200  =  60  x  500,  or  d  =  7.98  feet 


PRESSURE  TURBINE 


335 


The  forms  of  bucket  are  derived  from  the  velocity  diagrams.  For  the  first 
stage,  we  proceed  as  in  Art.  530,  using  the  relative  angles  e  and/" given  in  Fig.  255 
for  determining  the  angles  of  the  backs  of  the  moving  blades,  and  the  absolute 
angles  for  determining  those  of  the  stationary  blades. 

533.  Utilization  of  Pressure  Energy.  Besides  the  energy  of  impulse 
against  the  wheel,  unaccompanied  by  changes  in  pressure,  the  steam  may 
expand  while  traversing  the  buckets,  producing  work  by  reaction.  This 
involves  incomplete  expansion  in  the  nozzle,  and  makes  the  velocities  of 
the  discharged  jets  much  less  than  in  a  pure  impulse  turbine.  Lower 
rotative  speeds  are  therefore  practicable.  Loss  of  efficiency  is  avoided  by 
carrying  the  ultimate  expansion  down  to  the  condenser  pressure.  In  the 
pure  pressure  turbine  of  Parsons,  there  are  no  expanding  nozzles ;  all  of 
the  expansion  occurs  in  the  buckets  (Art.  524).  (See  Fig.  247.)  Here 
the  whole  useful  effort  is  produced  by  the  reaction  of  the  expanding  steam 
as  it  emerges  from  the  working  blades  to  the  guide  blades.  No  velocity  is 
given  up  during  the  passage  of  the  steam ;  the  velocity  is,  in  fact,  increasing, 
hence  the  name  reaction  turbine.  The  impulse  turbine,  on  the  contrary, 
performs  work  solely  because  of  the  force  with  which  the  swiftly  moving 
jet  strikes  the  vane.  It  is  sometimes  called  the  velocity  turbine.  Turbines 
are  further  classified  as  horizontal  or  vertical,  according  to  the  position  of 
the  shaft,  and  as  radial  flow  or  axial  flow,  according  to  the  location  of  the 
successive  rows  of  buckets.  Most  pressure  turbines  are  of  the  axial  flow 
type. 


534-  Design  of  Pressure  Turbine.  The  number  of  stages  is  now  large.  The 
heat  drop  in  any  stage  is  so  small  that  the  entering  velocity  is  no  longer  negligible. 
The  velocities  which  determine  the  rate  of  conversion  of  heat  into  work  will  vary 
during  the  passage  of  steam,  being  reduced  by  friction  and  increased  by  expansion : 
the  latter  being  provided  by  appropriately  shaping  the  buckets.  We  may  assume 

a  reduction  of  heat  drop  by  friction 
—  say  25  per  cent  —  and  plot  the  ex- 
pansive path  as  in  Fig.  240.  This 
permits  of  determining  the  pressure, 
volume,  and  quality  at  any  tempera- 
ture. 

In  Fig.  259,  let  the  turbine  have 
four  drums,  FC,  CD,  DB,  BO.  .The 
peripheral  speeds  of  these  drums  may 
vary  from  130  to  350  feet  per  second. 
We  will  now  assume  absolute  veloci- 
ties for  the  steam  entering  each  set  of  moving  blades,  as  along  EA.  .It  is  cus- 
tomary to  allow  these  velocities  to  range  from  1|  to  3|  times  the  peripheral  speed 
of  the  drums ;  they  should  increase  quite  rapidly  toward  the-  last  stages  of  ex- 
pansion. Knowing  the  steam  velocity  and  peripheral  velocity  for  any  state  like 


F 
FIG.  259. 


c 

Art.    534,    Prob.    17. —Design 
Pressure  Turbine. 


336 


APPLIED  THERMODYNAMICS 


Z,  we  construct  a  velocity  diagram  as  in  Fig.  249,  choosing  appropriate  angles  of 
entrance  and  exit.     In  ordinary  practice,  the  expansion  in  the  buckets  is  sufficient, 

notwithstanding  friction,  to  make  the  rela- 
tive exit  and  absolute  entrance  angles  and 
velocities  about  equal.  In  such  case,  we 
have  the  simple  graphical  construction  of 
Fig.  260. 

Since  ab  =  be,  db  =  be,  and  ad  =  ec,  we 
obtain 

,  _  u(ah  +  he)  _  ad(hc  +  hd) 
32.2  32.2 

Drop  the  perpendicular  bh,  and  with  h 
as  a  center  describe  the  arc  aj.  Draw 
dg  perpendicular  to  ac.  Then 


FIG.  260.    Art.  534,  Prob.  18.  — Velocity 
Diagram,  Pressure  Turbine. 


dg2  =  ad  x  dc  =  ad(dh  +  he),  and 


work  =     L.  foot-pounds,  or 
32.2 


B.  t.  u. 


In  the  general  case,  the  work  may  be  computed  as  in  Art.  532.  This  result 
represents  the  heat  converted  into  work  at  a  stage  located  vertically  in  line  with 
the  point  Z,  Fig.  259.  Let  this  heat  be  laid  off  to  some  convenient  scale,  as  GH. 
Similar  determinations  for  other  states  give  the  heat  drop  cur-ve  IJKHLMNOP. 
The  average  ordinate  of  this  curve  is  the  average  heat  drop  or  work  done  per 
stage.  If  we  divide  the  total  heat  drop  obtained  by  the  average  drop  per  stage, 
we  have  the  number  of  stages,  the  nearest  whole  number  being  taken.*  The 
diameter  of  any  drum  at  mid-height  of  buckets  is  computed  from  the  peripheral 
velocity  and  number  of  revolutions  per  minute. 

535.  Details.  The  bucket  spacing  and  heights  must  be  such  as  to  give  room 
for  the  passage  of  the  necessary  volume  of  steam,  which  depends  upon  the  turbine 
output  and  varies  with  the  stage  of  expansion  attained.  The  blade  heights  should 
be  at  least  3  per  cent  of  the  drum  diameter,  to  avoid  excessive  leakage  over  their 
tips.  The  clearance  over  tips  in  inches  should  be  from  0.01  d  to  0.008  d,  where  d 
is  the  drum  diameter  in  feet.  Blade  widths  vary  from  $  to  1^  in.,  with  center  to 
center  spacing  of  from  1£  to  4  in.  Blade  angles  are  obtained  from  the  velocity 
diagram. 

If  A  is  the  angle  made  between  the  steam  leaving  the  guide  vanes  and  the 
plane  of  the  wheel,  and  c  is  the  absolute  velocity  of  the  stream,  the  axial  com- 
ponent of  this  velocity  is  c  sin  A.  Let  the  number  of  buckets  on  a  wheel  (stage) 
be  n,  their  height  I,  and  their  spacing  e.  Without  allowance  for  thickness  of 
buckets,  the  area  for  passage  of  steam  would  be  nel\  the  usual  thickness  of 
buckets  will  reduce  this  to  $  nel.  The  volume  of  steam  discharged  per  second  will 
then  be  f  nelc  sin  A  —  Gw,  in  which  G  is  the  weight  of  flow  per  second  and  w  the 
specific  volume,  which  varies  while  the  steam  is  traversing  a  single  row  of  buckets. 


*  Dividing  the  total  heat  drop  at  a  state  in  a  vertical  line  through  C  by  the  average 
drop  per  stage  from  F  to  (7,  we  have  the  number  of  stages  on  the  first  drum. 


PRESSURE  TURBINE 


337 


Since  ne  is  the  circumference  of  the  wheel  =  ird,  where  d  is  the  diameter,  we  have 
|  irdlc  sin  A  =  Gw. 

The  successive  drum  diameters  frequently  have  the  ratio  V2 : 1  (21). 

Specimen  Case 

To  determine  the  general  characteristics  of  a  pressure  turbine  operating  be- 
tween pressures  of  100  and  3.5  lb.,  with  an  initial  superheat  of  300°  F.,  the  heat 
drop  being  reduced  25  per  cent  by  friction.  There  are  to  be  3  drums,  and  the  heat 
drop  is  to  be  equally  divided  between  the  drums.  The  peripheral  speeds  of  the 
successive  drums  are  160,  240,  320  ft  per  second.  The  relative  entrance  and 
absolute  exit  velocities  and  angles  are  equal :  the  absolute  entrance  angle  is  20°. 
The  turbine  makes  300  r.  p.  m.  and  develops  2500  kw.  with  losses  between  buckets 
and  generator  output  of  65  per  cent. 


1 

. 

•00 

v^ 

V 

j 

0 

4*0 

\ 

1325 

J_ 

\ 

tf 

1820 

1* 

«& 

\ 

'0 

1310 

^ 

\ 

'*C 

A 

vP 

\ 

'v, 

1290 

\ 

c/ 

V 

1280 

V 

\ 

\ 

-<<t 

? 

\ 

Vv 

^ 

A 

1250 

s 

f 

<• 

1 

S- 

'r 

1246  « 

14, 

A, 

3 

\ 

^ 

1230  § 

V 

1225  H 

-.«>. 

• 

•s 

1 

1^0  rj 

1220  *" 

\ 

\ 

I 

5 

• 

\ 

\v 

1210 

I 

^ 

1^ 

j  _S 

•y 

V- 

1205 

s 

1200 

ic 

oN 

V 

1196 

\ 

* 

N 

V 

1186 

>c 

\ 

1180 

\ 

' 

u- 

s 

,- 

1 

0  s 

^ 

1166 

s 

S 

01 

^  J* 

^  , 

4f? 

^ 

1160 

r°r 

N^ 

0 

1146 

S|-v 

s 

-; 

1130 

1126 

FIG.  2t>0  a.    Art.  535.  —  Expansion  Path,  Pressure  Turbine. 


338 


APPLIED  THERMODYNAMICS 


In  Fig.  260  a,  the  expansive  path  is  plotted  on  a  portion  of  the  total  heat- 
entropy  diagram.  The  total  heat  drop  is  shown  to  be  1342  -  1130  =  212  B.  t.  u., 
and  the  heat  drop  per  drum  is  212  -  3  =  70f  B.  t.  u.  In  Fig.  260  b,  lay  off  to  any 
scale  the  equal  distances  ab,  be,  cd,  and  the  vertical  distances  ae,  bg,  ci,  rep- 
resenting the  drum  speeds.  Lay  off  also  ak,  bm,  co,  equal  respectively  to 
1£  x  (ae,  bg,  ci),  and  al,  bn,  cp,  equal  respectively 
to  3^  times  these  drum  speeds.  The  curve  qr  p 


FIG.  260  b.    Art.  535.  —  Elements  of  Pressure  Turbine. 

of  entrance  absolute  velocities  is  now  assumed,  so  as  to  lie  wholly  within  the  area 
klsntpuvowmx.  Figure  260  c  shows  the  essential  parts  of  the  velocity  diagram 
for  the  stages  on  the  first  drum.  Here  ab  represents  aq  in  Fig.  260  b,  ad  represents 

=  3.12  B.  t.  u.  is  the  heat  drop 
for  the  first  stage  in  the  turbine.  Making  ac  represent  by  and  drawing  dc,  ch,  af, 
we  find  (T-T^-T  i  =  I  .  „..'.  l  =3.70  B.  t.  u.  as  the  heat  drop  for  the  last  stage  on 


ae,  the  angle  bad  is  20°,  and  f-^-V  =  f^ 

\lo8.3/        Xloo.o/ 


the  first  drum.     For  intermediate  stages  between  these  two,  we  find, 


INITIAL  ABSOLUTE 
VELOCITY 

ORDINATE  FROM 
d 

HEAT  DROP, 

B.  T.  U. 

ab  =  350 

de  =  279.7 

3.12 

356£ 

282.8 

3.20 

362| 

285.9 

3.26 

368| 

289.0 

3.34 

375 

292.1 

3.40 

381$ 

295.3 

3.48 

387| 

298.4 

3.56 

393| 

301.5 

3.63 

ac  =  400 

df=  304.7 

3.70 

PRESSURE  TURBINE 


339 


In  Fig.  260  b,  we  now  divide  the  distance  ab  into  8  equal  parts  and  lay  off  to 
any  convenient  vertical  scale  the  heat  drops  just  found,  obtaining  the  heat  drop 
curve  zA.  The  average  ordinate  of  this  curve  is  3.41  and  the  number  of  stages  on 
the  first  drum  is  70f  -f-  3.41  =  21  (nearest  whole  number).  The  number  of  stages 


FIG.  260  c.    Art.  535.  —  Velocity  Diagram,  Pressure  Turbine. 

on  the  other  drums  is  found  in  the  same  way,  the  peripheral  velocity  ad,  Fig. 
260  c,  being  different  for  the  different  drums.  The  diameter  d  of  the  first  drum  is 
given  by  the  expression 


300  vd  =  60  x  160  or  d  = 


60  x  160 


3.1416  x  300 
The  weight  of  steam  flowing  per  second  is 


=  10.2  ft. 


2500  x  1.34  x  2545 
0.65  x  212  x  3600 


=  17.1  Ib. 


340  APPLIED   THERMODYNAMICS 

In  the  first  stage  of  the  first  drum,  the  condition  of  the  steam  at  entrance  to 
the  guide  blades  is  (Fig.  260  a)  H  =  1342,  p  —  100;  at  exit  from  the  moving 
blades,  it  is  H  =  1338.59,  p  =  98.  From  the  total  heat-pressure  diagram,  or  by 
computation,  the  corresponding  specific  volumes  are  6.5  and  6.6.  The  volumes  of 
steam  flowing  are  then  6.5  x  17.1  =  111  and  6.6  x  17.1  =  113  cu.  ft.  per  second. 
The  absolute  steam  velocities  are  (Fig.  260  b)  350  and  356J  ft.  per  second.  The 
axial  components  of  these  velocities  (entrance  angle  20°)  are  0.34202  x  350  =  120, 
and  0.34202  x  356J-  =  122.  The  drum  periphery  is  10.2  x  3.1416  =  32  ft.  If  the 
blade  thicknesses  occupy  $  this  periphery  and  the  width  for  steam  passage  between 
the  buckets  is  constant,  the  width  for  passage  of  steam  isf  x  32  =  21.33  ft.  and 

the  necessary  height  of  fixed  buckets  is  —  —  =  0.434  ft.  or  5.2  in.  at  the 


beginning  of  the  stage  and  -  —  113         =  0.434  ft.  or  5.2  in.  at  the  end.     The 

fixed  blade  angles  are  determined  by  the  velocities  be  and  ab,  Fig.  260:  those  of 
the  moving  blades  by  bd  and  be.  There  is  no  serious  error  involved  in  taking  the 
velocity  and  specific  volume  as  constant  throughout  a  stage.  The  height  of  the 
moving  buckets  should  of  course  not  be.  less  than  that  of  the  guide  blades;  this 
may  be  accomplished  by  increasing  the  thickness  of  the  former. 

It  should  be  noted  that  the  velocities  indicated  by  the  curve  qr,  Fig.  260  &,  are 
those  of  the  steam  at  exit  from  the  fixed  blades  and  entrance  to  the  moving  blades. 
The  diagram  of  Fig.  260  gives  the  absolute  velocity  of  the  steam  entering  the  next 
set  of  fixed  blades. 


COMMERCIAL  FORMS  OF  TURBINE. 

536.  De  Laval;  Stumpf.  Figure  235  illustrates  the  principle  of  the  De  Laval 
machine,  the  working  parts  of  which  are  shown  in  Fig.  261.  Entering  through 
divergent  nozzles,  the  steam  strikes  the  buckets  around  the  periphery  of  the  wheel 
b.  The  shaft  c  transmits  power  through  the  helical  pinions  a,  a,  which  drive  the 
gears  e,  e,  e,  e,  on  the  working  shafts/,  f.  The  wheel  is  housed  with  the  iron  cas- 
ing g.  This  is  a  horizontal  single-stage  impulse  turbine,  with  a  single  wheel. 
Its  rotative  speed  is  consequently  high  ;  in  small  units,  it  reaches  30,000  r.  p.  m. 
It  is  built  principally  in  small  sizes,  from  5  to  300  h.  p.  The  nozzles  make  angles 
of  20°  with  the  plane  of  the  wheel ;  the  buckets  are  symmetrical,  and  their  angles 
range  from  32°  to  36°,  increasing  with  the  size  of  the  unit.  For  these  proportions, 
the  most  efficient  values  of  u  would  be  about  950  and  2100  for  absolute  steam  veloci- 
ties of  2000  and  4400  feet  per  second,  respectively ;  in  practice,  these  speeds  are 
not  attained,  u  ranging  from  500  to  1400  feet  per  second,  according  to  the  size. 
The  high  rotative  speeds  require  the  use  of  gearing  for  most  application?.  The 
helical  gears  used  are  quiet,  and  being  cut  right-  and  left-hand  respectively  they 
practically  eliminate  end  thrust  on  the  shaft.  The  speed  is  usually  reduced  in  the 
proportion  of  1  to  10.  The  high  rotative  speeds  also  prevent  satisfactory  balanc- 
ing, and  the  shaft  is,  therefore,  made  flexible  ;  for  a  5-hp.  turbine,  it  is  only  f 
inch  in  diameter.  The  bearings  h,  j  are  also  arranged  so  as  to  permit  of  some 
movement.  The  pressure  of  steam  in  the  wheel  case  is  that  of  the  atmosphere  or 
condenser,  all  expansion  occurring  in  the  nozzle.  A  centrifugal  governor  controls 


DE  LAVAL  TURBINE 


341 


342 


APPLIED  THERMODYNAMICS 


the  speed  by  throttling  the  steam  supply  and  by  opening  communication  between 
the  wheel  case  and  atmosphere  when  necessary. 

The  nozzles  of  the  De  Laval  turbine  are  located  as  in  Fig.  235.  Those  of  the 
Stumpf,  another  turbine  of  this  class,  are  tangential,  while  the  buckets  are  of  the 
Pelton  form  (Fig.  252),  and  are  milled  in  the  periphery  of  the  wheel.  A  very 
large  wheel  is  employed,  the  rotative  speeds  being  thus  reduced.  In  a  late  form 
of  the  Stumpf  machine,  a  second  stage  is  added.  The  reversals  of  direction  are  so 
.extreme  that  the  fluid  friction  must  be  excessive. 

537.  Curtis  Turbine.     This  is  a  multi-stage  impulse  turbine,  the  principle  of 
operation  having  been  shown  in  Fig.  245.     In  most  cases,  it  is  vertical ;  for  marine 

applications,  it  is  necessarily  made 
horizontal.  Figure  262  illustrates 
the  stationary  and  moving  blades 
and  nozzles.  Steam  enters  through 
the  nozzle  A ,  strikes  a  row  of  mov- 
ing vanes  at  a,  passes  from  them 
through  stationary  vanes  B  to 
another  row  of  moving  vanes  at  e, 
then  passes  through  a  second  set 
of  expanding  nozzles  at  h  to  the 
next  pressure  stage.  This  particu- 
lar machine  has  four  pressure 
stages  with  two  sets  of  moving 
buckets  in  each  stage.  The  direc- 
tion of  flow  is  axial.  The  number 
of  pressure  stages  may  range  from 
two  to  seven.  From  two  to  four 
velocity  stages  (rows  of  moving 
buckets)  are  used  in  each  pressure 
stage.  In  the  two-stage  machine, 
the  second  stage  is  disconnected 
when  the  turbine  runs  non-con- 
densing, the  exhaust  from  the  first 
stage  being  discharged  to  the  at- 
mosphere. Governing  is  effected 

by  automatically  varying  the  number  of  nozzles  in  use  for  admitting  steam  to  the 
first  stage.  A  step  bearing  carries  the  whole  weight  of  the  machine,  and  must  be 
supplied  with  lubricant  under  heavy  pressure ;  an  hydraulic  accumulator  system  is 
commonly  employed. 

538.  Rateau  Turbine.     This  is   a  horizontal,  axial  flow,  multi-stage  impulse 
turbine.     The  number  of  pressure  stages  is  very  large  — from  twenty-five  upward. 
There  is  one  velocity  stage  in  each  pressure  stage.     Very  low  speeds  are,  therefore, 
possible.     Figure  263  shows  the  general  arrangement ;  the  tranverse  partitions  e,  e 
form  cells,  in  which  revolve  the  wheels/,  /;  the  nozzles  are  merely  slots  in  the 
partitions.     The  blades  are  pressed  out  of  sheet  steel  and  riveted  to  the  wheel. 
The  wheels  themselves  are  of  thin  pressed  steel. 


FIG.  262.    Art.  537.  —  Curtis  Turbine. 


APPLICATIONS  OF  TURBINES 


343 


FIG.  263.    Art.  538.  -Rateau  Turbine. 

539.  Westinghouse-Parsons  Turbine.     This  is  of  the  axial  flow  pressure  type, 
and  horizontal.     The  steam  expands  through  a  large  number  of  successive  fixed 
and  moving  blades.     In  Fig.  264,  the  steam  enters  at  A  and  passes  along  the  vari- 
ous blades  toward  the  left ;  the  movable  buckets  are  mounted  on  the  three  drums, 
and  the  fixed  buckets  project  inward  from  the  casings.     The  diameters   of  the 
drums  increase  by  steps ;  the  increasing  volume  of  the  steam  within  any  section  is 
accommodated  by  varying  the  bucket  heights.     The  balance  pistons  P,  P,  P  are 
used  to  counteract  end  thrust.     The  speed  is  fairly  high,  and  special  provision 
must  be  made  for  it  in  the  design  of  the  bearings.     Governing  is  effected  by  inter- 
mittently opening  the  valve  V;  this  valve  is  wide  open  whenever  open  at  all. 

The  length  of  this  machine  is  sometimes  too  great  for  convenience.  To  over- 
come this,  the  "  double-flow  "  turbine  receives  steam  near  its  center,  through 
expanding  nozzles  which  supply  a  simple  Pelton  impulse  wheel.  This  utilizes 
a  large  proportion  of  the  energy,  and  the  steam  then  flows  in  both  directions 
axially,  through  a  series  of  fixed  and  moving  expanding  buckets.  Besides  reduc- 
ing the  length,  this  arrangement  practically  eliminates  end  thrust  and  the  neces- 
sity for  balance  pistons. 

540.  Applications  of  Turbines.     Turbo-locomotives  have  been  experimented 
with  in.  Germany;  the  direct  connection  of  the  steam  turbine  to  high-pressure 
rotary  air   compressors  has  been  accomplished.     In   stationary  work,  the  direct 
driving  of  generators  by  turbines  is  common,  and  the  high  rotative  speeds  of  the 
latter  have  cheapened  the  former.      At  high  speeds,  difficulties  may  be  experi- 
enced with  commutation ;  so  that  the  turbine  is  most  successful  with  alternating- 
current  machines.     When  driving  pumps,  turbines  permit  of  exceptionally  high 
lifts  with  good  efficiencies  for  the  centrifugal  type,  and  low  first  costs.     For  low- 
pressure,  high-speed  blowers,  the  turbine  is  an  ideal  motor.     The  outlook  for  a  gas 
turbine  is  not  promising,  any  gas  cycle  involving  combustion  at  constant  pressure 
being  both  practically  and  thermodynamically  inefficient. 

The  objections  to  the  turbine  in  marine  application  have  arisen  from  the  high 


344 


APPLIED  THERMODYNAMICS 


EXHAUST   STEAM   TURBINES  345 

speed  and  the  difficulty  of  reversing.  A  separate  reversing  wheel  may  be  em- 
ployed, and  graduation  of  speed  is  generally  attained  by  installing  turbines  in 
pairs.  A  small  reciprocating  engine  is  sometimes  employed  for  maneuvering  at 
or  near  docks.  Since  turbines  are  not  well  adapted  to  low  rotative  speeds,  they 
are  not  recommended  for  vessels  rated  under  15  or  16  knots.  The  advantages  of 
turbo-operation,  in  decreased  vibration,  greater  simplicity,  .smaller  and  more  deeply 
immersed  propellers,  lower  center  of  gravity  of  engine-room  machinery,  decreased 
size,  lower  first  cost,  and  greater  unit  capacity  without  excessive  size,  have  led  to 
extended  marine  application.  The  most  conspicuous  examples  are  in  the  Cunard 
liners  Lusitania  and  Mauretania.  The  former  has  two  high-pressure  and  two  low- 
pressure  main  turbines,  and  two  astern  turbines,  all  of  the  Parsons  type  (22). 
The  drum  diameters  are  respectively  96, 140,  and  104  in.  An  output  of  70,000  hp. 
is  attained  at  full  speed. 

541.  The  Exhaust-steam  Turbine.  From  the  heat  chart,  Fig.  177,  it  is 
obvious  that  steam  expanding  adiabatically  from  150  Ib.  absolute  pressure  and 
600°  F.  to  1.0  Ib.  absolute  pressure  transforms  into  work  365  B.  t.  u.  It  has  been 
shown  that  in  the  ordinary  reciprocating  engine  such  complete  expansion  is  unde- 
sirable, on  account  of  condensation  losses.  The  final  pressure  is  rarely  below  7  Ib. 
absolute,  at  which  the  heat  converted  into  work  in  the  above  illustration  is  only 
252  B.  t.  u.  The  turbine  is  particularly  fitted  to  utilize  the  remaining  113  B.  t.  u. 
of  available  heat.  The  use  of  low-pressure  turbines  to  receive  the  exhaust  steam 
from  reciprocating  engines,  has,  therefore,  been  suggested.  Some  progress  has 
been  made  in  applying  this  principle  in  plants  where  the  engine  load  is  intermit- 
tent and  condensation  of  the  exhaust  would  scarcely  pay.  With  steel  mill  en- 
gines, steam  hammers,  and  similar  equipment,  the  introduction  of  a  low-pressure 
turbine  is  decidedly  profitable.  The  variations  in  supply  of  steam  to  the  turbine 
are  offset  by  the  use  of  a  regenerator  or  accumulator,  a  cast-iron,  water-sprayed 
chamber  having  a  large  storage  capacity,  constituting  a  "  fly  wheel  for  heat,"  and 
by  admitting  live  steam  to  the  turbine  through  a  reducing  valve.  When  a  sur- 
plus of  steam  reaches  the  accumulator,  the  pressure  rises ;  as  soon  as  this  falls, 
some  of  the  water  is  evaporated.  The  maximum  pressure  is  kept  low  to  avoid 
back  pressure  at  the  engines.  A  steam  consumption  by  the  turbine  as  low  as 
35  Ib.  per  brake  hp.-hr.  lias  been  claimed,  with  15  Ib.  initial  absolute  pressure  and 
a  final  vacuum  of  26  in.  Other  good  results  have  been  shown  in  various  trials 
(23).  Wait  (24)  has  described  a  plant  at  South  Chicago,  111.,  in  which  a  42  by 
60  double  cylinder,  reversible  rolling-mill  engine  exhausts  to  an  accumulator  at  a 
pressure  2  or  3  Ib.  above  that  of  the  atmosphere.  This  delivers  steam  at  about 
atmospheric  pressure  to  a  500  kw.  Rateau  turbine  operated  with  a  28-in.  vacuum. 
The  steam  consumption  of  the  turbine  was  about  35  Ib.  per  electrical  hp.-hr., 
delivered  at  the  switchboard. 

The  S.S.  Turbinia,  in  1897,  was  fitted  with  low-pressure  turbines  receiving  the 
exhaust  from  reciprocating  engines  and  operating  between  9  Ib.  and  1  Ib.  absolute. 
One  third  of  the  total  power  of  the  vessel  was  developed  by  the  turbines,  although 
the  initial  pressure  was  160  Ib. 

542.  Commercial  Considerations.  The  best  turbines,  in  spite  of  their  thermo- 
dynamically  superior  cycle,  have  not  yet  equalled  in  efficiency  the  best  reciprocat- 


346  APPLIED  THERMODYNAMICS 

• 

ing  engines,  both  operating  at  full  load.  The  average  turbine  is  more  economical 
than  the  average  engine ;  and  since  the  mechanical  and  fluid  friction  losses  are 
disproportionately  large,  it  seems  reasonable  to  expect  improved  efficiencies  as 
experimental  knowledge  accumulates. 

The  turbine  is  cheaper  than  the  engine ;  it  weighs  less,  has  no  fly  wheel, 
requires  less  space  and  very  much  less  foundation.  It  can  be  built  in  larger  units 
than  a  reciprocating  cylinder.  Power  house  buildings  are  cheapened  by  its  use  ;  the 
cost  of  attendance  and  of  sundry  operating  supplies  is  reduced.  It  probably  depre- 
ciates less  rapidly  than  the  engine.  The  wide  range  of  expansion  makes  a  high 
vacuum  desirable  ;  this  leads  to  excessive  cost  of  condensing  apparatus.  Similarly, 
superheat  is  so  thoroughly  beneficial  in  reducing  steam  friction  losses  that  a  con- 
siderable investment  in  superheaters  is  necessary.  The  turbine  must  have  a  direct 
connected  balanced  load  ;  so  that  the  cost  of  generators  must  often  be  included  in 
the  initial  expense,  although  otherwise  unnecessary.  The  choice  as  between  the 
turbine  and  the  engine  must  be  determined  with  reference  to  all  of  the  condi- 
tions, technical  and  commercial,  including  that  of  load  factor.  Turbine  economy 
cannot  be  measured  by  the  indicator,  but  must  be  determined  at  the  brake  or 
switchboard  and  should  be  expressed  on  the  heat  unit  basis  (B.  t.  u.  consumed  per 
unit  of  output  per  minute). 

(1)  Trans.  Inst.  Engrs.  and  Shipbuilders  in  Scotland,  XLVI,  V.  (2)  Berry, 
The  Temperature-Entropy  Diagram,  1905.  (3)  To  show  this,  put  the  expression  in 

v 

the  brace  equal  to  m,  and  make  —  =0  :  then  —  =  I y  +     )       ,  which  may  be  solved 

dp  p      \    2     J 

for  any  given  value  of  y.  (4)  Thesis,  Polytechnic  Institute  of  Brooklyn,  1905. 
(5)  Thomas,  Steam  Turbines,  1906,  89.  (6)  Proc.  Inst.  Civ.  Eng.,  CXL,  199. 
(7)  Zeits.  Ver.  Deutsch.  Ing.,  Jan.  16,  1904.  (8)  Rankine,  The  Steam  Engine,  1897, 
344.  (9)  Experimental  Researches  on  the  Flow  of  Steam,  Brydon  tr. ;  Thomas,  op.  cit., 
106.  (10)  Thomas,  op.  cit.,  123.  (11)  Engineering,  XIII  (1872).  (12)  Trans. 
A.  S.  M.  E.,  XI,  187.  (13)  Mitteil.  ilber  Forschungsarb.,  XVIII,  47.  (14)  Practice 
and  Theory  of  the  Injector,  1894.  (15)  Peabody,  Thermodynamics,  1907,  443. 
(16)  Trans.  A.  S.  M.  E.,  XXVII,  081.  (17)  Stodola,  Steam  Turbines.  (18)  The 
Steam  Engine,  1905,  I,  170.  (19)  Technical  Thermodynamics,  Klein  tr.,  1907:  I, 
225:  II,  153.  (20)  Trans.  A.  S.  M.  E.,  XXVII,  081.  (21)  See  H.  F.  Schmidt,  in 
The  Engineer  (Chicago),  Dec.  16,  1907:  Trans.  Inst.  Engrs.  and  Shipbuilders  in 
Scotland,  XLXIX.  (22)  Power,  November,  1907,  770.  (23)  Trans.  A.  S.  M.  E., 
XXV,  817 :  Ibid,  XXXII,  3,  315.  (24)  Proc.  A.  I.  E.  E.,  1907. 

OUTLINE   OF  CHAPTER   XIV 

The  turbine  utilizes  the  velocity  energy  of  a  jet  or  stream  of  steam. 

Expansion  in  a  nozzle  is  adiabatic,  but  not  isentropic  ;  the  losses  in  a  turbine  are  due 

to  residual  velocity,  friction  of  steam  through  nozzles  and  buckets  and  mechanical 

friction. 

E  +  PW+  ^  =  e  +  pw  +  ~,  or  1J  =  q -  Q,  approximately  ; 
Z  y  Zg          2  g 

whence  V=  223.84  Vq^~Q. 

The  complete  expansion  secured  in  the  turbine  warrants  the  use  of  exceptionally  high 
vacuum. 


THE  STEAM   TURBINE  347 

Nozzle  friction  decreases  the  heat  converted  into  work  and  the  velocity  attained  ; 

V=  212.42  Vq^~Q. 
The  heat  expended  in  overcoming  friction  reappears  in  drying  or  superheating  the 

steam. 

W  T* 

F  =  G  — ,  which  reaches  a  minimum  at  a  definite  vahie  of  —  •     For  steam,  this  value 

is  about  0.57.    If  the  discharge  pressure  is  less  than  0.57  p,  the  nozzle  converges  to 

a  "throat"  and  afterward  diverges. 

The  multi-stage  impulse  turbine  uses  lower  rotative  speeds  than  the  single  stage. 
The  diverging  sides  of  the  nozzle  form  an  angle  of  10°  ;  the  converging  portion  may  be 

one  fourth  as  long. 

Steam  consumption  per  Ihp.-hr.  =  2545  -4-  E(c[  —  $). 
The  rotative  components  of  the  absolute  velocities  determine  the  work ;  the  relative 

velocities  determine  the  (moving)  bucket  angles.    Bucket  friction  may  decrease 

relative  velocities  by  10  per  cent  during  passage.     Work  =  (v  cos  a  ±  Ycosg")  -. 

9 

Efficiency  =  E  =  Work  -5-  778  (q  —  $).    Bucket  angles  may  be  adjusted  to  equalize 

end  thrust,  to  secure  maximum  work,  or  may  be  made  equal. 
For  a  right-angled  stream  change,  maximum  efficiency  is  0.50 ;  with  complete  reversal, 

it  is  1.00.     With  practicable  buckets,  it  is  always  less  than  1.0. 
The  backs  of  moving  buckets  are  made  tangent  to  the  relative  stream  velocities. 
The  angles  of  fixed  blades  are  determined  by  the  absolute  velocities. 
In  the  pure  pressure  turbine,  expansion  occurs  in  the  buckets.     No  nozzles  are  used. 
Turbines  may  be  horizontal  or  vertical,  radial  or  ax,ial  flow,  impulse  or  pressure  type. 

In  designing  a  pressure  turbine,  -  =  0.30  to  0.75.     The  heat  drop  at  any  stage  may 

v 

equal  (  -38L  ]  ?  Fig.  260.    The  number  of  stages  is  the  quotient  of  the  whole  heat 

V 158. 3/ 

drop,  corrected  for  friction,  by  the  mean  value  of  this  quantity.    Friction  through 
buckets  may  be  from  20  to  30  per  cent.    The  accumulated  heat  drop  to  any  stage 
is  ascertained  and  the  condition  of  the  steam  found  as  in  Fig.  240. 
Commercial  forms  include  the  De  Laval,  single-stage  impulse  : 

Stumpf,  single-  or  two-stage  impulse,  with  Pelton  buckets. 
Curtis,  multi-stage  impulse,  usually  vertical,  axial  flow. 
Bateau,  multi-stage  impulse,  axial  flow,  horizontal,  many  stages. 
Westinghouse-Parsons,  pressure  type,  axial  flow,  horizontal ;  sometimes  of  the 
"  double  flow''  form. 

Marine  applications  involve  some  difficulty,  but  have  been  satisfactory  at  high  speeds. 
The  turbine  may  utilize  economically  the  heat  rejected  by  a  reciprocating  engine.     A 

regenerator  is  sometimes  employed. 

The  best  recorded  thermal  economy  has  been  attained  by  the  reciprocating  engine ; 
but  commercially  the  turbine  has  many  points  of  superiority. 

PROBLEMS 

1.  Show  on  the  TJV  diagram  the  ideal  cycle  for  a  turbine  operating  between  pressure 
limits  of  140  Ib.  and  2  lb.,  with  an  initial  temperature  of  500°  F.  and  adiabatic 
(isen tropic)  expansion.  What  is  the  efficiency  of  this  cycle  ? 


348  APPLIED  THERMODYNAMICS 

2.  In  Problem  1,  what  is  the  loss  of  heat  contents  and  the  velocity  ideally  attained  ? 

3.  In  Problem  1,  how  will  the  efficiency  and  velocity  be  affected  if  the  initial 
pressure  is  150  Ib  ?  If  the  initial  temperature  is  600°  F.  ?  If  the  final  pressure  is  1  Ib.  ? 

4.  Solve  Problems  1,  2,  and  3,  making  allowance  for  friction  as  in  Art.  519. 

5.  Compute  analytically,  in  Problem  3,  first  case,  the  condition  of  the  steam  after 
expansion,  as  in  Art.  520,  assuming  the  heat  drop  to  have  been  decreased  10  per  cent 
by  friction. 

6.  An  ideal  reciprocating  engine  receives  steam  at  150  Ib.  pressure  and  550°  F., 
and  expands  it  adiabatically  to  7  Ib.  pressure.     By  what  percentage  would  the  efficiency 
be  increased  if  the  steam  were  afterward  expanded  adiabatically  in  a  turbine  to  1.5  Ib. 
pressure  ? 

7.  Steam  at  100  Ib.  pressure,  92  per  cent  dry,  expands  to  16  Ib.  pressure.     The  loss 
of  heat  drop  due  to  friction  is  10  per  cent.     Compute  the  final  condition  and  the  velocity 
attained. 

8.  In  Problem  5,  find  the  throat  and  outlet  diameters  of  a  nozzle  to  discharge 
1000  Ib.  of  steam  per  hour,  and  sketch  the  nozzle. 

T> 

9.  Check  the  value  -  =  0.5274  for  maximum  flow  in  Art.  522. 

P 

10.  Check  the  equation  of  flow  of  a  permanent  gas,  in  Art.  522. 

11.  If  the  efficiency  in  Problem  5,  from  steam  to  shaft,  is  0.60,  find  the  steam  con- 
sumption per  brake  hp.-hr  ,  and  the  thermal  efficiency. 

12.  In  Problem  5,  let  the  peripheral  speed  be  u  =  480,  the  angle  a  =  20°,  and  find 
the  work  done  per  pound  of  steam  in  a  single-stage  impulse  turbine  (a)  with  end  thrust 
eliminated,  (ft)  with  equal  relative. angles.     Allow  a  10  per  cent  reduction  of  relative 
velocity  for  bucket  friction. 

13.  In  Problem    12,  Case  (&),  what  is  the  efficiency  from  steam  to  work  at  the 
buekets?     (Item  E,  Art.  526.) 

14.  Sketch  the  bucket  in  Problem  12,  Case  (6),  as  in  Art.  530. 

15.  Compute  the  wheel  diameters  and  design  the  first-stage  nozzles  and  buckets  for 
a  two-stage  impulse  turbine,  with  two  moving  wheel.-}  in  each  stage,  as  in  Art.  532, 
operating  under  the  conditions  of  Problem  5,  the  capacity  to  be  1500  kw.,  the  entering 
stream  angles  20°,  the  peripheral  speed  600  ft.  per  second,  the  speed  1500  r.  p.  m.,  the 
heat  drop  reduced  0.10  by  nozzle  friction.     Arrange  the  bucket  angles  to  give  the  highest 
practicable  efficiency,*  the  stream  velocities  to  be  reduced  10  percent  by  bucket  friction. 
State  the  heat  unit-consumption  per  kw. -minute. 

16.  In  Problem  5,  plot  by  stages  of  about  10  B.  t.  u.  the  NT  expansion  path  in  a 
pressure  turbine  in  which  the  heat  drop  is  decreased  0.25  by  bucket  friction. 

17.  In  Problem  16,  the  drums  have  peripheral  speeds  of  150,  250,  350.     Construct  a 
reasonable  curve  of  steam  velocities,  as  in  Fig.  259,  the  velocity  of  the  steam  entering 
the  first  stage  being  400  ft.  per  second,  and  the  corrected  heat  drop  through  the  drums 
being  equally  divided. 

18.  In  Problem  17,  let  the  absolute  entrance  angles  be  20°,  and  let  the  velocity 
diagram  be  as  in  Fig.  260.     Find  the  work  done  in  each  of  six  stages  along  each  drum. 
Find  the  average  heat  drop  per  stage,  and  the  number  of  stages  in  each  drum,  the  total 
heat  drop  per  drum  having  been  obtained  from  Problem  16. 

*The  angle /must  not  be  less  than  24°  in  any  case. 


THE  STEAM   TURBINE  349 

19.  The  speed  of  the  turbine  in  Problem  18  is  400  r.  p.  m.     Find  the  diameter  of  each 
drum. 

20.  In  Problems  16-19,  the  blades  are  spaced  2"  centers.     The  turbine  develops 
1500  kw.    Find  the  heights  of  the  moving  blades  for  one  expansive  state,  assuming 
losses  between  buckets  and  generator  of  45  per  cent.     Design  the  moving  bucket. 

21.  Sketch  the  arrangement  of  a  turbine  in  which  the  steam  first  strikes  a  Pelton 
impulse  wheel  and  then  divides;  one  portion  traveling  through  a  three-drum  pressure 
rotor  axially,  the  other  through  a  two-pressure  stage  velocity  rotor  with  three  rows  of 
moving  buckets  in  each  pressure  stage,  also  axially,  the  shaft  of  the  velocity  turbine 
being  vertical. 

22.  Compare  as  to  effect  on  thermal   efficiency  the    methods   of  governing   the 
De  Laval,  Curtis,  and  Westinghouse-Parsons  turbines. 

23.  Determine  whether  the  result  given  in  Art.  541,  reported  for  the  S.S.  Turbinia, 
is  credible. 


CHAPTER   XV 

RESULTS  OF  TRIALS  OF  STEAM  ENGINES  AND  STEAM  TURBINES 

543.  Sources.     The  most  reliable  original  sources  of  information  as  to  con- 
temporaneous steam  economy  are  the  Transactions  or  Proceedings  of  the  various 
national  mechanical  engineering  societies  (1).     The  reports  of  the  Committee  of 
the  Institution  of  Mechanical  Engineers  on  Marine  Engine  Trials  are  of  special 
interest  (2).     The  Alsatian  experiments  on  superheating  have  already  been  re- 
ferred to  (Art.  443).     The  works  of  Barrus  (3)  and  of  Thomas  (4)  present  a  mass 
of  results  obtained  on   reciprocating  engines   and   turbines  respectively.     The 
investigations  of  Isherwood  are  still  studied  (5). 

544.  Limiting  Efficiencies.     Neither  the  engine  nor  the  turbine  can,  in  prac- 
tice, give  an  efficiency  equal  to  that  of  the  corresponding  Clausius  cycle.     Actual 
tests  show  efficiency  ratios  ranging  usually  between  50  and  75  per  cent,  but  occa- 
sionally overlapping  one  of  these  limits.     The  Clausius  efficiency  depends  solely 
upon   the   temperature  limits,  so  that  we  may  expect  engine   efficiencies   to  be 
improved  by  high  pressures  or  superheats  and  good  vacua.     The  actual  engine  is 
subject  to  various  additional  modifying  conditions ;  in  general,  for  a  given  tem- 
perature range  in  the  cylinder,  we  may  find  the  efficiency  to  be  improved  by  well- 
designed   valves,  fairly  low   terminal   pressures  and  reasonably  wide  ratios   of 
expansion,  jackets  (unless  the  steam   is  superheated),  and   multiple   expansion. 
Since  engines  are  generally  governed  by  varying  the  ratio  of  expansion,  we  may 
find  that  steady  loads,  as  in  pumping  service,  which  lead  to  uniform  ratios  of 
expansion,  are  also  associated  with  maximum  efficiencies. 

545.  Basis  for  Rating.     The  heat  unit  basis  is  the  only  proper  standard  for 
comparing  the  performance  of  engines  operating  under  dissimilar  conditions.     On 
account  of  the  uncertainty  which  has  existed  as  to  the  specific  heat  of  superheated 
steam,  various  constant  or  variable  values  have  been  employed  in  computing  the 
results  of  trials  in  which  superheat  was  used.     These  lead  to  results  not  strictly 
comparable,  although  the  error  can  seldom  be  of   much  consequence.     For  the 
present,  at  least,  trials  made  with  superheated  steam  should  be  so  reported  that 
the  correction  for  superheat  may  be  independently  made  by  any  one,  using  such 
values  as  he  prefers  for  the  specific  heat. 

546.  Non-condensing  Trials.     Usual   steam  rates    (pounds  of   dry  steam  per 
Ihp.-hr.)  range  from  21.5  (with  jackets)  up  to  38,  in  good  simple  engines,  when 
new.     A  fair  rate  with  an  unjacketed  cylinder  is  30 ;  poor  engines,  such  as  direct- 

350 


RESULTS   OF  TRIALS 


351 


acting  steam  pumps,  without  expansion,  show  steam  rates  up  to  319  pounds  (6) 
or  more.  Dean  (7)  quotes  a  number  of  tests  on  high-speed  single-valve  and  four- 
valve  engines,  showing  that  the  best  efficiency  is  obtained  at  rather  low  ratios  of 
expansion,  and  that  the  economy  falls  off  rapidly  with  wear.  Mechanical  efficiencies 
range  from  75  to  90  per  cent;  the  combined  mechanical  efficiency  of  a  small  direct- 
connected  engine  and  generator  of  these  types  may  be  taken  under  ordinary  con- 
ditions at  75  per  cent.  Steam  pressures  seldom  range  above  100  lb.* 
PI  Multiple-expansion  engines  seldom  run  non-condensing. 

Willans  found  for  compounds  steam  rates  of  19.14  to  23 ;  for 
triples  a  minimum  rate  of  18.5  has  been  obtained  (8). 

547.  Simple  Condensing  Engines.     As  early  as  1840,  the 
famous   Cornish  pumping  engines,  with  expansion  ratios  from 
1.5  to  3.5,  gave,  under  the  best  conditions,  steam  rates  of  16.5 
to  24  lb.    (9).     No  improvement  has  been  made  over  these 
figures ;  usual  rates  range  from   16.9  (with   jackets)  to  24.2. 
The  famous  Leavitt  pumping  engine  at  Lawrence  gave  16.5, 
with  120  lb.  initial  pressure,  16  expansions,  and  12  r.  p.  m. 

548.  Compound  Condensing  Engines.     Steam   rates   range 
downward  from  21  lb.,  in  good  types  well  operated.     In  1878, 
the  Corliss  Pawtucket  pumping  engine  gave  13.7  lb.  with  120  lb. 
steam  pressure.     Pressures  now  range  up  to  175  lb.     Rock- 
wood's  high  ratio  compound  (Art.  480),  with  150  lb.  steam 
pressure,  gave  a  rate  of  12.45.     Jacobus  (10)  tested  a  Rice  and 
Sargent  engine  which  gave  12.10  lb.     This  ran,  at  150  lb.  steam 
pressure  and  28  inches  of  vacuum,  at  120  r.  p.  m.     The  com- 
bined diagrams  showing  the  effect  of  the  jackets  and  reheaters 
appear  in  Fig.  265.      A  curve  showing  the  steam  consumption 
at  various  loads  is  given  in  Fig.  266.     A  heat  unit  consumption 
of  222  B.  t.  u.  per  Ihp.-minute  was  reached  at  normal  load  of 
700  hp. :   the  economy  held  up  well  at  heavy  overloads,  a  point 

of  much  com- 
mercial impor- 
tance. A250hp. 
Van  den  Ker- 
chove  engine 
(11)  at  126 
r.  p.  m.,  130  lb. 
pressure,  and 
32  expansions 
gave  a  rate  of 

11.98  lb.  A  Westinghouse  engine  of  5400  hp.  at  185  lb.  pressure  gave  11.93  lb. 
Barrus  and  Rockwood,  with  175  lb.  pressure,  obtained  the  best  rate  thus  far  re- 
ported —  11.22  lb.  —  on  another  "  wide  ratio  "  compound. 


FIG.  265.    Art.  548.  — Rice  and  Sargent  Engine  Diagrams. 


*  Pressures  given  in  this  chapter,  unless  otherwise  specified,  are  gauge  pressures. 


352 


APPLIED   THERMODYNAMICS 


549.  Triple-Expansion  Condensing  Engines.  The  experimental  engine  at 
the  Massachusetts  Institute  of  Technology  gave  the  following  heat  unit  consump- 
tions per  Ihp.  per  minute :  319  as  a  compound  without  jackets ;  274  as  a  triple 
without  jackets  ;  261  with  jackets  on  heads ;  239  \\ith  jackets  on  whole  of  cylinders 


15 

\ 

\ 

N 

13 

X 

"\ 

s. 

•>- 

,^—  -  ' 

.  — 

•s" 

1 

1 

j 

1 

1 

I 

\ 

! 

i 

\ 

4 

1 

i, 

g 

! 

1 

INDICATED  HORSE  POWER 

FIG.  266.    Art.  548.  — Test  of  Rice  and  Sargent  Engine. 

and  receiver ;  233  with  jackets  on  cylinders  only.  Steam  rates,  in  practice,  range 
downward  from  16  lb.,  with  steam  pressures  usually  under  200  Ib.  Willans  ob- 
tained 12.74  lb. ;  Schroter,  12.2  and  12.65  lb.  A  very  small  engine  has  given 
12.68  lb.  (12).  A  rate  of  12.5  lb.  would  be  extremely  good  in  ordinary  mill 
service.  With  only  124  lb.  pressure,  and  a  merely  fair  vacuum,  Cooley  (13)  ob- 
tained 12.65  lb.  on  a  15,000,000-gallon  Nordberg  pumping  engine.  At  an  even 
lower  pressure  (121.4  lb.),  an  Allis  pumping  engine  gave  an  11.68  lb.  rate. 
Laird  reports  (14)  for  two  10,000,000-gallon  Allis  pumps  at  136.5  lb.  pressure,  an 
average  rate  of  11.63  lb.  or  216.7  B.  t.  u.  per  Ihp.  per  minute,  with  a  mechanical 
efficiency  of  94.6  per  cent.  A  20,000,000-gallon  Snow  pump  tested  by  Goss  (15) 
with  155  lb.  pressure,  gave  a  rate  of  11.38  lb.  and  94  per  cent  mechanical  effi- 
ciency. The  Leavitt  engine  gave  11.22  lb.  with  176  lb.  steam  pressure.  The  best 
rate  recorded  at  its  date  on  saturated  steam  was  made  by  the  Allis  pumping 
engine  at  Hackensack,  N.J.,  about  1904.  This  used  a  pressure  of  188  lb.,  33 
expansions,  and  ran  at  30  r.  p.  m. ;  its  steam  rate  was  11.05  lb.,  or  211  B.  t.  u. 
per  Ihp.  per  minute.  The  best  triple  has  thus  only  slightly  excelled  the  best 
compounds. 


550.    Quadruple   Engines. 


N  9   h 

FIG.  267.    Art.  550.  — Nordberg  Engine 
Diagrams. 


The  most  economical  performances  on  record 
with  saturated  steam  have  been  made 
in  quadruple-expansion  engines.  The 
Nordberg  pumping  engine  at  Wildwood 
(16)  although  of  only  6,000,000  gal. 
capacity  (712  horse  power),  and  jack- 
eted on  barrels  of  cylinders  only,  gave 
a  heat  consumption  of  185.96  B.  t.  u. 
with  200  lb.  initial  pressure  and  only  a 
fair  vacuum.  The  high  efficiency  was 
obtained  by  drawing  off  live  steam 
from  each  of  the  receivers  and  trans- 
ferring its  high-temperature  heat  direct 
to  the  boiler  feed  water  by  means  of 
coil  heaters.  Heat  was  thus  absorbed 
more  nearly  at  the  high  temperature 


TESTS   WITH   SUPERHEATED   STEAM 


353 


256.78 

"THROTTLE 


limit,  and  a  closer  approach  made  to  the  Carnot  cycle  than  in  the  ordinary  en- 
gine.    Thus,  in  Fig.  267,  BCDS  represents  the  Clausius  cycle.     The  heat  areas 
hIHE,  gKJh,  NMLg  represent  the  withdrawal  of  steam  from  the 
various  receivers,  these  amounts  of  heat  being  applied  to  heating 
the  water  along  Bd,  de,  ef.    The  heat  imparted  from  icithout  is  then 
only  cfCDE.     The  work  area  DHIJKLMRS  has  been  lost,  but 
the  much  greater  heat  area  ABfc  has  been  saved,  so  that  the  effi- 
ciency is  increased.     The  cycle  is  regenerative ;  if  the  number  of 
steps  were  infinite,  the  expansive  path  would  be  DF,  parallel  to 
BC,  and  the  cycle  would  be  equally  efficient  with  that  of  Carnot. 
The  actual  efficiency  was  68  per  cent  of  that  of  the  Carnot  cycle. 
The  steam  rate  was  not  low,  being  increased  by  the  system  of 
drawing  off  steam  for  the  heaters  from  11.4  to  12.26;  but  the  real 
efficiency  was,  at  the  time,  unsurpassed.     A  later  test  of  a  Xord- 
berg   engine  of   similar   construction,  used  to   drive  an  air  com- 
pressor, is  reported  by  Hood  (17).     Here  the  combined  diagrams 
were  as  in  Fig.  268.     Steam  was  received  at  257  Ib.  pressure,  the 
vacuum   being   rather   poor.     At  normal  capacity,  1000  hp.,  the 
mechanical  efficiency  was  90.35  per  cent,  and  the  heat  consump- 
tion  169.29  B.  t.  u.      This 
appears  to  be  the  best  record 
to  date.      The  efficiency  is 
73.69  per  cent  of  that  of  the 
Carnot  cycle,  and  88.2  per 
cent  of  that  of  the  Clausius 
cycle. 


1.24  CONDENSER 


FIG.  268.    Art.  550.  —  Hood  Compressor  Diagrams. 


551.  Superheated  Steam ;  Reciprocating  Engines.  At  150  Ib.  pressure  and 
250°  of  superheat,  Schroter  obtained  heat  rates  of  199  to  223  B.  t.  u.  with  super- 
heated steam,  against  213  to  246  B.  t.  u.  with  saturated  steam  in  the  same  engine, 
the  gain  by  superheating  being  greater  at  wide  ranges  of  expansion.  Jacobus 
(18)  found  on  a  small  compound  Rice  and  Sargent  engine,  a  steam  rate  of  9.56 
Ib.  when  about  400°  of  superheat  was  used,  with  a  rate  of  13.84  Ib.  for  saturated 
steam.  The  pressure  was  140  Ib.  and  the  vacuum  only  fair.  The  engine  was, 
however,  poorly  adapted  for  the  use  of  saturated  steam.  A  result  of  exceptional 
interest  was  obtained  in  Carpenter's  tests  (19)  of  the  engines  of  the  White  steam 
motor  car.  The  maximum  output  was  only  45  hp.,  the  weight  of  the  entire  power 
plant  only  643  Ib.  The  engine  was  cross-compound,  running  condensing.  The 
boiler  pressure  ranged  up  to  595  Ib.,  with  as  much  of  300°  of  superheat ;  the 
exhaust  from  the  engine  was,  in  fact,  superheated.  A  steam  rate  as  low  as  10.8 
Ib.  was  obtained,  or  of  12  Ib.  per  brake  horse  power,  corresponding  to  246  B.  t.  u. 
per  brake  horse  power  per  minute.  The  Van  den  Kerchove  engine  mentioned  in 
Art.  548  gave,  with  superheat,  a  steam  rate  of  8.99,  and  a  heat  unit  consumption 
of  192  B.  t.  u. 


552.    Turbines.     With  pressures  of  from  78.8  to  140  Ib.,  and  vacuum  from 
24.3  to  26.4  in.,  steam  rates  per  brake  horse  power  of  18.0  to  23.2  Ib.  have  been 


354  APPLIED  THERMODYNAMICS 

obtained  with  saturated  steam  on  De  Laval  turbines.  Dean  and  Main  (20)  found 
corresponding  rates  of  15.17  to  16.54  with  saturated  steam  at  200  Ib.  pressure,  and 
13.94  to  15.62  with  this  steam  superheated  91°. 

Parsons  turbines,  with  saturated  steam,  have  given  rates  per  brake  horse 
power  from  14.1  to  18.2;  with  superheated  steam,  from  12.6  to  14.9.  This  was  at 
120  Ib.  pressure.  A  7500-kw.  unit  tested  by  Sparrow  (21)  with  177.5  Ib.  initial 
pressure,  95.74°  of  superheat,  and  27  in.  of  vacuum,  gave  15.15  Ib.  of  steam  per 
kw.-hr.  The  Stott  engine-turbine  outfit  (see  footnote  23,  Chapter  XIV)  gave  a 
thermal  efficiency  of  0.206  from  steam  to  generator  output  while  the  load  varied 
from  6500  kw.  to  15,500  kw.  Bell  reports  for  the  Lusitania  (22)  a  coal  consump- 
tion of  1.43  Ib.  per  horse  power  delivered  at  the  shaft.  Denton  quotes  (23)  10.28 
Ib.  per  brake  horse  power  on  a  4000  hp.  unit,  with  190°  of  superheat  (214  B.  t.  u.  per 
minute);  and  13.08  on  a  1500-hp.  unit  using  saturated  steam.  A  400-kvv.  unit 
gave  11.2  Ib.  with  180°  of  superheat.  A  1250-kw.  turbine  gave  13.5  Ib.  with 
saturated  steam,  12.8  with  100°  of  superheat,  13.25  with  77°  of  superheat  (24). 
(All  per  brake  hp.-hr.) 

A  Rateau  machine,  with  slight  superheat,  gave  rates  from  15.2  to  19.0  Ib.  per 
brake  horse  power.'  Curtis  turbines  have  shown  14.8  to  18.5  Ib.  per  kw.-hr.,  as  the 
superheat  decreased  from  230°  to  zero,  and  of  17.8  to  22.3  Ib.  as  the  back  pressure 
increased  from  0.8  to  2.8  Ib.  absolute.  Kruesi  has  claimed  (25)  for  a  5000-kw. 
Curtis  unit,  with  125°  of  superheat,  a  steam  rate  of  14  Ib.  per  kw.-hr. ;  and  for  a 
2000-kw.  unit,  under  similar  conditions,  16.4  Ib. 

553.  Summary.  The  following  table  represents  the  best  results  as  above 
given,  with  some  of  the  results  to  be  expected  in  ordinary  practice  with  usual 
good  engines  operating  at  reasonably  steady  loads : 

SATURATED  STEAM 
TYPE  OF  ENGINE  BEST  STEAM  RATE  AVERAGE  STEAM  RATE 

IHP.  IHP. 

Simple,  Non-Condensing                         21.5  38.0 

Compound,  Non-Condensing                   19.14  23.0 

Simple,  Condensing                                 16.5  22.0 

Compound,  Condensing                            11.22  18.0 
Triple,  Condensing                                    11.05 

Quadruple,  Condensing  (169.29  B.  t.  u.)  

.  BHP. 

Single  Stage  Velocity  Turbine               15.17 
Pressure  Turbine  13.08  

SUPERHEATED  STEAM 
IHP. 

Compound,  Condensing  8.99  (192  B.  t.  u.)  

BHP. 
Single  Stage  Velocity  Turbine  13.94  

Pressure  Turbine  ] 

,,  ,, •    ,         ,7  T     .,     rr     l  .          approximately  10  

Multi -stage  Velocity  Turbine    ] 


ENGINE   FRICTION 


355 


554.  Locomotive  Tests.  The  surprisingly  low  steam  rate  of  16.60  Ib.  has 
been  obtained  at  200  Ib.  pressure,  with  superheat  up  to  192°.  This  is  equivalent 
to  a  rate  of  17.8  Ib.  with  saturated  steam.  The  tests  at  the  Louisiana  Purchase 
Exposition  (26)  showed  an  average  steam  rate  of  20.23  Ib.  for  all  classes  of  engines 
tested,  or  of  21.97  for  simple  engines  and  18.55  for  compounds,  with  steam  pres- 
sures ranging  from  200  to  225  Ib.  These  results  compare  most  favorably  with  any 
obtained  from  high-speed  non-condensing  stationary  engines.  The  mechanical 
efficiency  of  the  locomotive,  in  spite  of  its  large  number  of  journals,  is  high ;  in 
the  tests  referred  to,  under  good  conditions,  it  averaged  88.3  per  cent  for  consoli- 
dation engines  and  89.1  per  cent  for  the  Atlantic  type.  The  reason  for  these  high 
efficiencies  arises  from  the  heavy  overload  carried  in  the  cylinder  in  ordinary  ser- 
vice. The  maximum  equivalent  evaporation  per  square  foot  of  heating  surface 
varied  from  8.55  to  16.34  Ib.  at  full  load,  against  a  usual  rate  not  exceeding  4.0  Ib. 
in  stationary  boilers ;  the  boiler  efficiency  consequently  was  low,  the  equivalent 
evaporation  per  pound  of  dry  coal  (14,000  B.  t.  u.)  falling  from  11.73  as  a  maxi- 
mum to  6.73  as  a  minimum,  between  the  extreme  ranges  of  load.  Notwithstand- 
ing this,  a  coal  consumption  of  2.27  Ib.  per  Ihp.-hr.  has  been  reached.  These  trials 
were,  of  course,  laboratory  tests;  road  tests,  reported  by  Hitchcock  (27),  show  less 
favorable  results ;  but  the  locomotive  is  nevertheless  a  highly  economical  engine, 
considering  the  conditions  under  which  it  runs. 


555.  Engine  Friction.  Excepting  in  the  case  of  turbines,  the  figures  given 
refer  usually  to  indicated  horse  power,  or  horse  power  developed  by  the  steam  in 
the  cylinder.  The  effective  horse  power,  exerted  by  the  shaft,  or  brake  horse 
power,  is  always  less  than  this,  by  an  amount  depending  upon  the  friction  of  the 
engine.  The  ratio  of  the  latter  to  the  former  gives  the  mechanical  efficiency,  which 
may  range  from  0.85  to  0.90  in  good  practice  with  rotative  engines  of  moderate 
size,  and  up  to  0.965  in  exceptional  cases.  The  brake  horse  power  is  usually  deter- 
mined by  measuring  the  pull  exerted  on  a  friction  brake  applied  to  the  belt  wheel. 
When  an  engine  drives  a  generator,  the  power  indicated  in  the  cylinder  may  be 
compared  with  that  developed  by  the  generator,  and  an  over-all  efficiency  of 
mechanism  thus  obtained.  The  difficulties  involved 
have  led  to  the  general  custom,  in  turbine  practice,  of 
reporting  steam  rates  per  kw.-hr.  Thnrston  has  em-  " 
ployed  the  method  of  driving  the  engine  as  a  ma-  5 
chine  from  some  external  motor,  and  measuring  the  s 
power  required  by  a  transmission  dynamometer. 

In  direct-driven  pumps,  air  compressors,  and  re- 
frigerating machines,  the  combined  mechanical  effi- 
ciency  is  found  by  comparing  the  indicator  diagrams 
of  the  steam  and  pump  cylinders.  These  efficiencies 
are  high,  on  account  of  the  decrease  in  number  of 
bearings,  crank  pins,  and  crosshead  pins. 


556.    Variation    in    Friction.      Theoretically,    at   FlG> 
least,    the   friction    includes   two    parts  :    the    initial 


Art    556.  -Engine 
Friction. 


356 


APPLIED   THERMODYNAMICS 


—600 


_400 


friction,  that  of  the  stuffing  boxes,  which  remains  practically  constant ;  and  the 
load  friction,  of  guides,  pins,  and  bearings,  which  varies  with  the  initial  pressure 

and  expansive  ratio.  By  plotting 
concurrent  values  of  the  brake  horse 
power  and  friction  horse  power,  we 
thus  obtain  such  a  diagram  as  that 
of  Fig.  269,  in  which  the  height  ab 
represents  the  constant  initial  fric- 
tion, and  the  variable  ordinate  xy 
the  load  friction,  increasing  in  arith- 
metical proportion  with  the  load. 
It  has  been  found,  however,  that  in 
practice  the  total  friction  is  more 
affected  by  accidental  variations  in 
lubrication,  etc.,  than  by  changes  in 
load,  and  that  it  may  be  regarded  as 
practically  constant,  for  a  given  en- 


_300 


_200 


— 1PC 


INDICATED  HORSE  POWER 

FIG.  270.    Art.  556.  — Willans  Line  for  Constant 
Initial  Pressure. 

gine,  at  all  loads. 

The  total  steam  consumption  of  an  engine  at  any  load  may  then  be  regarded 
as  made  up  of  two  parts  :  a  constant  amount,  necessary  to  overcome  friction  ;  and 
a  variable  amount,  necessary  to 
do  external  work,  and  varying 
with  the  amount  of  that  work. 
AVillans  found  that  this  latter 
part  varied  in  exact  arithmeti- 
cal proportion  with  the  load, 
when  the  output  of  the  engine 
was  varied  by  changing  the  initial 
pressure;  a  condition  repre- 
sented by  the  Willans  line  of 
Fig.  270  (28).  The  steam  rate 
was  thus  the  same  for  all  loads, 
excepting  as  modified  by  fric- 
tion. (Theoretically,  this 
should  not  hold,  since  lowering 
of  the  initial  pressure  lowers 
the  efficiency.)  When  the  load 


TOW 

2200 

/ 

/ 

/\ 

1600 
1100 

^/ 

/ 

•# 

y 

2 

xi 

800- 

/ 

7 

400 

10   20   3 

0   40   50   C 

0  70   £ 

0   90  100  110  120 

ELECTRICAL  HORSE  POWER 


FIG.  271.    Art.   556,  Prob.   16.— Willans  Line  for  a 

Parsons  Turbine. 
is  changed  by  varying  the  ratio 

of  expansion,  the  corrected  steam  rate  tends  to  decrease  with  increasing  ratios, 
and  to  increase  on  account  of  increased  condensation ;  there  is,  however,  some 
gain  up  to  a  certain  limit,  and  the  Willans  line  must,  therefore,  be  concave  up- 
ward. Figure  271  shows  the  practically  straight  line  obtained  from  a  series  of 
tests  of  a  Parsons  turbine.  If  the  line  for  an  ordinary  engine  were  perfectly 
straight,  with  varying  ratios  of  expansion,  the  indication  would  be  that  the  gain 
by  more  complete  expansion  was  exactly  offset  by  the  increase  in  cylinder  con- 
densation. A  jacketed  engine,  in  which  the  influence  of  condensation  is  largely 
eliminated,  should  show  a  maximum  curvature  of  the  Willans  line. 


MECHANICAL   EFFICIENCY 


357 


557.  Variation  in  Mechanical  Efficiency.  With  a  constant  friction  loss,  the 
mechanical  efficiency  must  increase  as  the  load  increases,  hence  the  desirability 
of  running  engines  at  full  capacity.  This  is  strikingly  illustrated  in  the  locomo- 
tive (Art.  554).  Engines  operating  at  serious  variations  in  load,  as  in  street  rail- 
way power  plants,  may  be  quite  wasteful  on  account  of  the  low  mean  mechanical 
efficiency.  A  secondary  effect  enters  here,  on  account  of  the  rapidity  of  fluctuation 
of  the  load ;  this  leads  to  losses  both  mechanical  and  thermodynamic,  which, 
although  of  importance,  have  never  been  satisfactorily  analyzed. 


Art     ^S.-Vngine  Friction 
and  Limit  of  Expansion. 


558.  Limit  of  Expansion.     Aside  from  cylinder  condensation,  engine  friction 
imposes  a  limit  to  the  desirable  range  of  expansion.     Thus,  in  Fig.  272,  the  line 
ab  may   be  drawn  such   that  the   constant 

pressure  a  represents  that  necessary  to  over- 
come the  friction  of  the  engine.  If  ex- 
pansion is  carried  below  ab,  say  to  c,  the 
force  exerted  by  the  steam  along  be  will  be 
less  than  the  resisting  force  of  the  engine, 
and  will  be  without  value.  The  maximum 
desirable  expansion,  irrespective  of  cylinder 
condensation,  is  reached  at  b. 

559.  Distribution  of  Friction.     Experi- 
menting in   the   manner  described  in  Art. 
555,  Thurston  ascertained   the  distribution 
of  the  friction  load  by  successively  remov- 
ing  various  parts  of  the  engine  mechanism. 
Extended   tests   of   this    nature,    made    by 

Carpenter  and  Preston  (29)  indicate  that  from  35  to  47  per  cent  of  the  whole 
friction  load  is  imposed  by  the  shaft  bearings,  from  22  to  49  per  cent  by  the  piston, 
piston  rod,  pins,  and  slides  (the  greater  part  of  this  arising  from  the  piston  and 
rod),  and  the  remaining  load  by  the  valve  mechanism. 

(1)  Trans.  A.  S.  M.  E.,  Proc.  Inst.  JM".  E.,  Zeits.  Ver.  Deutsch.  Ing.,  etc.  (See 
The  Engineering  Digest,  November,  1908,  p.  542.)  (2)  Proc.  Inst.  Mech.  Eng.,  from  1889. 
(3)  Engine  Tests,  by  Geo.  H.  Barrus.  (4)  Steam  Turbines,  1906,  208-267.  (5)  Re- 
searches in  Experimental  Steam  Engineering.  (6)  Peabody,  Thermodynamics,  1907, 
244;  White,  Jour.  Am.  Soc.  Xav.  Engrs.,  X.  (7)  Trans.  A.  S.  M.  E.,  XXX,  6,  811. 
(8)  Ewing,  The  Steam  Engine,  1906,  177.  (9)  Denton,  The  Stevens  Institute  Indi- 
cator, January,  1905.  (10)  Trans.  A.  S.  M.  E.,  XXIV,  1274.  (11)  Denton,  op.  cit. 
'(12)  Ewing,  op.  cit.,  180.  (13)  Trans.  A.  S.  M.  E.,  XXI,  1018.  (14)  Ibid.,  XXI,  327. 
(15)  Ibid.,  XXI,  793.  (16)  Ibid.,  XXI,  181.  (17)  Ibid.,  XXVIII,  2,  221.  (18)  Ibid., 
XXV,  264.  (19)  Ibid.,  XXVIII,  2,  225.  (20)  Thomas,  Steam  Turbines,  1906,  212. 
(21)  Power,  November,  1907,  p.  772.  (22)  Proc.  Inst.  Nov.  Archts.,  April  9,  1908. 
(23)  Op.  cit.  (24)  Trans.  A.  S.  M.  E.,  XXV,  745  et  seq.  (25)  Power,  December, 
1907.  (26)  Locomotive  Tests  and  Exhibits,  published  by  the  Pennsylvania  Railroad. 
(27)  Trans.  A.  S.  M.  E.,  XXVI,  054.  (28)  Min.  Proc.  Inst.  C.  E.,  CXIV,  1893. 
(29)  Peabody,  op.  cit.,  p.  296. 


358  APPLIED  THERMODYNAMICS 

PROBLEMS 

(See  footnote,  Art.  546.) 

1.  Find  the  efficiency  ratio  (thermal  efficiency  as  compared  with  that  of*  the 
Carnot  cycle),  for  the  best  compound  engine  in  Art.  548,  if  the  vacuum  was  27  in.,* 
the  steam  as  received  being  dry. 

2.  Find  the  heat  unit  consumption  of  an  engine  using  30  Ib.  of  dry  steam  per 
Ihp.-hr.  at  100  Ib.  gauge  pressure,  discharging  this  steam  at  atmospheric  pressure.   How 
much  of  the  heat  (ignoring  radiation  losses)  in  each  pound  of  steam   is  rejected  ? 
What  is  the  quality  of  the  steam  at  release  ? 

3.  What  is  the  mechanical  efficiency  of  an  engine  developing  300  Ihp.,  if  30  hp. 
is  employed  in  overcoming  friction  ? 

4.  Why  is  it  unprofitable  to  run  multiple  expansion  engines  non-condensing  ? 

5.  Check  the  heat  unit  consumption  given  for  the  Rice  and  Sargent  engine  in 
Art.  548,  and  find  how  much  it  increased  at  20  per  cent  overload. 

6.  Make  deductions  from  Art.  549  as   to   the  value   of  triple   expansion   and 
jacketing. 

7.  Check  all  of  the  efficiency  ratios  given  in  Art.  550,  assuming  a  vacuum  of  26 
in.  in  each  case.     Explain  the  low  heat  unit  consumption  in  spite  of  the  high  steam 
rate. 

8.  Find  the  heat  unit  consumption  with  superheat  for  the  Rice  and  Sargent 
engine  in  Art.  551,  if  the  vacuum  was  27  in. 

9.  What  would  have  been  the  thermal  efficiency  of  the  White  motor  car  engine 
in  Art.  551  if  the  Carnot  efficiency  ratio  had  been  equal  to  that  of  the  Hood  compressor 
(Art.  550)  ?     (The  temperature  of  saturated  steam  at  695  Ib.  gauge  pressure  is  489°  F.) 
Compare  with  the  gas  engine  figures  in  Art.  343. 

10.  Find  the  heat  unit  consumptions  corresponding  to  the  figures  in  Art.  552  for 
De  Laval  turbines,  assuming  the  vacuum  to  have  been  27  in. 

11.  Find  the  heat  unit  consumption  for  the  7500  kw.  unit  in  Art.  552. 

12.  Estimate  the  probable  limit  of  boiler  efficiency  of  the  plant  on  the   S.S. 
Lusitania,  if  the  coal  contained  14,200  B.  t.  u.  per  Ib. 

13.  Determine  from  data  given  in  Art.  554  whether  a  coal  consumption  of  2.27 
Ib.  per  Ihp.-hr  is  credible  for  a  locomotive. 

14.  Using  values  given  for  the  coal  consumption  and  mechanical  efficiency,  with 
.how  little  coal  (14,000  B.  t.  u.  per  pound),  might  a  locomotive  travel  100  miles  at  a  speed 

of  50  miles  per  hour,  while  exerting  a  pull  at  the  drawbar  of  22,000  Ib.  ?  Make  compari- 
sons with  Problem  8,  Chapter  II,  and  determine  the  possible  efficiency  from  coal  to 
drawbar. 

*  Vacua  are  measured  downward  from  atmospheric  pressure.  One  atmosphere  = 
14.696  Ib.  per  square  inch  —  —  29.921  inches  of  (mercury)  vacuum.  If  p  =  absolute 
pressure,  pounds  per  square  inch,  v  =  vacuum  in  inches  of  mercury, 

\      .  14.696/ 
(29.921  -  v). 


PROBLEMS  359 

15.  An  engine  consumes  220  B.  t.  u.  per  Ihp.-min.,  360  B.  t.  u.  per  kw.-min.  of 
generator  output.    The  generator  efficiency  is  0.93.     What  is  the  mechanical  efficiency 
of  the  direct-connected  unit  ? 

16.  From  Fig.  271,  plot  a  curve  showing  the  variation  in  steam  consumption  per 
hp.-hr.  as  the  load  changes. 

17.  An  engine  works  between  150  and  2  Ib.  absolute  pressure,  the  mechanical 
efficiency  being  0.75.     What  is  the  desirable  ratio  of  (hyberbolic)  expansion,  friction 
losses  alone  being  considered  ? 

18.  If  the  mechanical  efficiency  of  a  rotative  engine  is  0.85,  what  should  be  its 
mechanical  efficiency  when  directly  driving  an  air  compressor,  based  on  the  minimum 
values  of  Art.  559  ? 


CHAPTER   XVI 

THE  STEAM  POWER  PLANT 

560.  Fuels.     The  complex  details  of  steam  plant  management  arise 
largely  from  differences   in   the  physical   and   chemical  constitution  of 
fuels.      "  Hard  "  coal,  for  example,  is  compact  and  hard,  while  soft  coal  is 
friable ;  the  latter  readily  breaks  up  into  small  particles,  while  the  former 
maintains  its  initial  form  unless  subjected  to  great  intensity  of  draft. 
Hard  coal,  therefore,  requires  more  draft,  and  even  then  burns  much  less 
rapidly  than  soft  coal ;  and  its  low  rate  of  combustion  leads  to  important 
modifications  in  boiler  design  and  operation  in  cases  where  it  is  to  be  used. 
Soft  coal  contains  large  quantities  of  volatile  hydrocarbons ;  these  distill 
from  the  coal  at  low  temperature,  but  will  not  remain  ignited  unless  the 
temperature  is  kept  high  and  an  ample  quantity  of  air  is  supplied.     The 
smaller  sizes  of  anthracite  coal  are  now  the  cheapest  of  fuels,  in  propor- 
tion to  their  heating  value,  along  the  northern  Atlantic  seaboard ;  but  the 
supply  is  limited  and  the  cost  increasing.     In  large  city  plants,  where 
fixed  charges  are  high,  soft  coal  is  often  commercially  cheaper  on  account 
of  its  higher  normal  rate  of  combustion,  and  the  consequently  reduced 
amount  of  boiler  surface  necessary.     The  sacrifice  of  fuel  economy  in 
order  to  secure  commercial  economy  with  low  load  factors  is  strikingly 
exemplified   in  the  "double  grate"  boilers  of  the  Philadelphia  Rapid 
Transit  Company  and  the  Interborough  Rapid  Transit  Company  of  New 
York  (1). 

561.  Heating  Value.     The  heating  value  of  a  fuel  is  determined  by  completely 
burning  it  in  a  calorimeter,  and  noting  the  rise  in  temperature  of  the  calorimeter 
water.     The  result  stated  is  the  number  of  heat  units  evolved  per  pound  with 
products  of  combustion  cooled  down  to  32°  F.     Fuel  oil  has  a  heating  value 
upward  of  18,000  B.  t.  u.  per  pound ;  its  price  is  too  high,  in  most  sections  of  the 
country,  for  it  to  compete  with  coal.     Wood  is  in  some  sections  available  at  low 
cost ;  its'  heating  value  ranges  from  6500  to  8500  B.  t.  u.     The  heating  values  of 
commercial  coals  range  from  8800  to  15,000  B.  t.  u.     Specially  designed  furnaces 
are  usually  necessary  to  burn  wood  economically. 

562.  Boiler  Room  Engineering.  While  the  limit  of  progress  in  steam  engine 
economy  has  apparently  been  almost  realized,  large  opportunities  for  improvement 
are  offered  in  boiler  operation.  This  is  usually  committed  to  cheap  labor,  with 

360 


EFFICIENCY  OF  COMBUSTION 


361 


insufficient  supervision.  Proper  boiler  operation  can  often  cheapen  power  to  a 
greater  extent  than  can  the  substitution  of  a  good  engine  for  a  poor  one.  New 
designs  and  new  test  records  are  not  necessary.  Efficiencies  already  reported 
equal  any  that  can  be  expected;  but  the  attainment  of  these  efficiencies  in  ordi- 
nary operation  is  essential  to  the  continuance  in  use  of  steam  as  a  power  produc- 
ing medium. 

563.  Combustion.     One  pound  of  pure  carbon  burned  in  air  uses  2.67 
Ib.  of  oxygen,  forming  a  gas  consisting  of  3.67  Ib.  of  carbon  dioxide  and 
8.90  Ib.  of  nitrogen. 

If  insufficient  air 
is  supplied,  the 
amount  of  carbon 
dioxide  decreases, 
some  carbon  mon- 
oxide being 
formed.  If  the  air 
supply  is  50  per  § 
cent,  deficient,  no 
carbon  dioxide  can 
(theoretically  at 
least)  be  formed. 
With  air  in  excess, 
additional  free 
oxygen  and  nitro- 
gen will  be  found 
in  the  products  of  combustion.  Figure  273  illustrates  the  percentage 
composition  by  weight  of  the  gases  formed  by  combustion  of  pure  carbon 
in  varying  amounts  of  air.  The  proportion  of  carbon  dioxide  reaches  a 
maximum  when  the  air  supply  is  just  right. 

564.  Temperature  Rise.     In  burning  to  carbon  dioxide,  each  pound  of 
carbon  evolves  14,500  B.  t.  u.     In  burning  to  carbon  monoxide,  only  4450 
B.  t.  u.  are  evolved  per  pound.     Let  W  be  the  weight  of  gas  formed  per 
pound  of  carbon,  K  its  mean  specific  heat,  T  —  t  the  elevation  of  tempera- 

14500 
ture  produced ;  then  for  combustion  to  carbon  dioxide,  T  —  t  =  —      -  and 


oogg 


oooo 


AIR  SUPPLY.«PERCENTAGE  OF  AM'T  THEOR.  NECESSARY  FOR  COMBUSTION 

FIG.  273.    Arts.  563,  564.  —  Air  Supply  and  Combustion. 


for  combustion  to  carbon  monoxide,   T  —  t  — 


4450 
WK 


WK 

The  rise  of  tempera- 


ture is  much  less  in  the  latter  case.  As  air  is  supplied  in  excess,  W 
increases  while  the  other  quantities  on  the  right-hand  sides  of  these  equa- 
tions remain  constant,  so  that  the  temperature  rise  similarly  decreases. 
The  temperature  elevations  are  plotted  in  Fig.  273.  The  maximum  rise 
of  temperature  occurs  when  the  air  supply  is  just  the  theoretical  amount. 


362  APPLIED  THERMODYNAMICS 

565.  Practical  Modifications.  These  curves  truly  represent  the  phe- 
nomena of  combustion  only  when  the  reactions  are  perfect.  In  practice, 
the  fuel  and  air  are  somewhat  imperfectly  mixed,  so  that  we  commonly 
find  traces  of  free  oxygen  and  carbon  monoxide  along  with  carbon  dioxide. 
The  best  results  are  obtained  by  supplying  some  excess  of  air ;  instead  of 
the  theoretical  11.57  lb.,  about  16  Ib.  may  be  supplied,  in  good  practice. 
In  poorly  operated  plants,  the  air  supply  may  easily  run  up  to  50  or  even 
100  lb.,  the  percentage  of  carbon  dioxide,  of  course,  steadily  decreasing. 
Gases  containing  10  per  cent,  of  dioxide  by  volume  are  usually  considered 
to  represent  fair  operation. 

566.  Distribution  of  Heat.     Of  the  heat  supplied  to  the  boiler  by  the  fuel, 
ignoring  radiation  losses,  a  part  is  employed  in  making  steam,  a  small  amount  of 
fuel  is  lost  through  the  grate  bars,  some  heat  is  transferred  to  the  external  atmos- 
phere, and  some  is  carried  away  by  the  heated  gases  leaving  the  boiler.      This 
last  is  the  important  item  of  loss.     Its  amount  depends  upon  the  weight  of  gases, 
their  specific  heat  and  temperature.     The  last  factor  we  aim  to  fix  in  the  design 
of  the  boiler  to  suit  the  specific  rate  of  combustion  :  the  specific  heat  we  cannot 
control;  but  the  weight  of  gas  is  determined  solely  by  the  supply  of  air,  and  is  sub- 
ject to  operating  control. 

Efficient  operation  involves  the  minimum  possible  air  supply  in 
excess  of  the  theoretical  requirement;  it  is  evidenced  by  the  per- 
centage of  carbon  dioxide  in  the  discharged  gases.  If  the  air  supply 
be  too  much  decreased,  however,  combustion  may  be  incomplete, 
forming  carbon  monoxide,  and  another  serious  loss  will  be  experienced, 
due  to  the  potential  heat  carried  off  by  the  gas. 

567.  Air  Supply  and  Draft.     The  draft  necessary  is  determined  by  the  physical 
nature  of  the  fuel ;  the  air  supply,  by  its  chemical  composition.     The  two  are  not 
equivalent ;  soft  coal,  for  example,  requires  little  draft,  but  ample  air  supply.     The 
two  should  be  subject  to  separate  regulation.     Low  grade  anthracite  requires  ample 
draft,  but  the  air  supply  should  be  closely  economized.     If  forced  draft,  by  steam 
jet,  blower,  or  exhauster,  is  employed,  the  necessary  head  should  be  provided  with- 
out the  delivery  of  an  excessive  quantity  of  air. 

568.  Types  of  Boiler.     Boilers  are  broadly  grouped  as  fire-tube  or  water-tube, 
internally  or  externally  fired.     A  type  of  externally  fired  water4<ibe  boiler  has  been 
shown  in  Fig.  233.     In  this,  the  Babcock  and  Wilcox  design,  the  path  of  the  gases 
is  as  described  in  Art.  508.     The  feed  water  enters  the  drum  6  at  29,  flows  down- 
ward through  the  back  water  legs  at  a,  and  then  upward  to  the  right  along  the 
tubes,  the  high  temperature  zone  at  1  compelling  the  water  above  it  in  the  tubes 
to  rise.     Figure  274  shows  the  horizontal  tubular  boiler,  probably  most  generally 
used  in  this  country.     The  fire  grate  is  at  S.     The  gases  pass  over  the  bridge  wall 
0,  under  the  shell  of  the  boiler,  up  the  back  end  Y,  and  to  the  right  through  tubes 


STEAM  BOILERS 


363 


running  from  end  to  end  of  the  cylindrical  shell.  The  tubes  terminate  at  C,  and 
the  gases  pass  up  and  away.  Feed  water  enters  the  front  head,  is  carried  in  the 
pipe  about  two  thirds  of  the  distance  to  the  back  end,  and  then  falls,  a  compensating 


HH— - T 

ttftfl 


upward  current  being  generated  over  the  grate.  This  is  an  externally  fired  fire-tube 
boiler.  Figure  275  shows  the  well-known  locomotive  boiler,  which  is  internally  fired. 
The  coldest  part  of  this  boiler  is  at  the  end  farthest  from  the  grate,  on  the  exposed 
sides.  The  feed  is  consequently  admitted  here.  Figure  276  shows  a  boiler  com- 
monly used  in  marine  service.  The  grate  is  placed  in  an  internal  furnace  ;  the 
gases  pass  upward  in  the  back  end,  and  return  through  the  tubes.  The  feed  pipe 
is  located  as  in  horizontal  tubular  boilers. 


364 


APPLIED  THERMODYNAMICS 


569.    Discussion.     The  internally  fired  boiler  requires  no  brick  furnace 

setting,  and  is  compact. 
The  water- tube  boiler  is 
rather  safer  than  the  fire- 
tube,  and  requires  less 
space.  It  can  be  more 
readily  used  with  high 
steam  pressures.  The  im- 
portant points  to  observe 
in  boiler  types  are  the 
paths  of  the  gases  and  of 
the  water.  The  gases 
should,  for  economy,  im- 
pinge upon  and  thoroughly 
circulate  about  all  parts 
of  the  heating  surface; 
the  circulation  of  the 
water  for  safety  and  large 
capacity  should  be  posi- 
tive and  rapid,  and  the 
cold  feed  water  should  be 
introduced  at  such  a  point 
as  to  assist  this  circula- 
tion. 

There  is  no  such  thing 
as  a  "most  economical 
type"  of  boiler.  Any 
type  may  be  economical 
if  the  proportions  are 
right.  The  grade  of  fuel 
used  and  the  draft  attain- 
able determine  the  neces- 
sary area  of  grate  for  a 
given  fuel  consumption. 
The  heating  surface  must 
be  sufficient  to  absorb  the 
heat  liberated  by  the  fuel. 
The  higher  the  rate  of 
combustion  (pounds  of  fuel 
burned  per  square  foot  of 
grate  per  hour),  the  greater 
the  relative  amount  of 
heating  surface  necessary. 


STEAM   BOILER  ECONOMY 


365 


LOMGJTUDINAL  SECTION 

FIG.  270.     Art.  568.  —  Marine  Boiler.     (Tbe  Bigelow  Company.) 

Rates  of  combustion  range  from  12  Ib.  with  low  grade  hard  coal  and 
natural  draft  up  to  30  or  40  Ib.  with  soft  coal ;  *  the  corresponding  ratios 
of  heating  surface  to  grate  surface  may  vary  from  30  up  to  60  or  70. 
The  best  economy  has  usually  been  associated  with  high  ratios.  The 
rate  of  evaporation  is  the  number  of  pounds  of  water  evaporated  per 
square  foot  of  heating  surface  per  hour;  it  ranges  from  3.0  upward,  de- 
pending upon  the  activity  of  circulation  of  water  and  gases.  An  effective 
heating  surface  usually  leads  to  a  low  flue-gas  temperature  and  relatively 
small  loss  to  the  stack.  Small  tubes  increase  the  efficiency  of  the  heat- 
ing surface  but  may  be  objectionable  with  certain  fuels.  Tubes  seldom 
exceed  20  ft.  in  length.  In  water-tube  boilers,  the  arrangement  of  tubes 
is  important.  If  the  bank  of  tubes  is  comparatively  wide  and  shallow, 
the  gases  may  pass  off  without  giving  up  the  proper  proportion  of  their 
heat.  If  the  bank  is  made  too  high  and  narrow,  the  grate  area  may  be 
too  much  restricted.  The  gases  must  not  be  allowed  to  reach  the  flue  too 
quickly. 

570.    Boiler  Capacity.     A  boiler  evaporating  3450  Ib.  of  water  per  hour 
from  and  at  212°  F.  performs  970  x  778  x  3450  =  2,600,000,000  foot-pounds 

*  Much  higher  rates  are  attained  in  locomotive  practice;  and  in  torpedo  boats,  with 
intense  draft,  as  much  as  200  Ib.  of  coal  may  be  burned  per  square  foot  of  grate  per  hour. 


366  APPLIED  THERMODYNAMICS 

of  work,  or  1300  horse  power.  No  engine  can  develop  this  amount  of  power 
from  3450  Ib.  of  steam  per  hour ;  the  power  developed  by  the  engine  is 
very  much  less  than  that  by  the  boiler  which  supplies  it.  Hence  the  custom 
of  rating  boilers  arbitrarily.  By  definition  of  the  American  Society  of 
Mechanical  Engineers,  a  boiler  horse  power  means  the  evaporation  of  344  Ib. 
of  water  per  hour  from  and  at  212°  F.  This  rating  was  based  on  the 
assumption  (true  in  1876,  when  the  original^  definition  was  established) 
that  an  ordinary  good  engine  required  about-  this  amount  of  steam  per 
horse  power  hour.  This  evaporation  involves  the  liberation  of  about 
33,000  B.  t.  u.  per  hour. 

571.  Limit  of  Efficiency.     The  gases  cannot  leave  the  boiler  at  a 
lower  temperature  than  that  of  the  steam  in  the  boiler.     Let  t  be  the 
initial  temperature  of  the  fuel  and  air,  x  the  temperature  of  the  steam, 
and  T  the  temperature  attained  by  combustion ;  then  if  W  be  the 
weight  of  gas  and  K  its  specific  heat,  assumed  constant,  the  total 
heat  generated  is  WK(T  —  £),  the  maximum  that  can  be  utilized  is 
WK(T  —  #),  and  the  limit  of  efficiency  is 

T-x 

T-t' 

In  practice,  we  have  as  usual  limiting  values  5T=4850,  #=350,  £=60  ; 
whence  the  efficiency  is  0.94  — a  value  never  reached  in  practice. 

572.  Boiler  Trials.     A  standard  code  for  conducting  boiler  trials  has 
been  published  by  the  American  Society  of  Mechanical   Engineers    (2). 
A  boiler,  like  any  mechanical  device,  should  be  judged  by  the  ratio  of  the 
work  which  it  does  to  the  energy  it  uses.     This  involves  measuring  the 
fuel  supplied,  determining  its  heating  value,  measuring  the  water  evaporated, 
and  the  pressure,  superheat,  or  wetness  of  the  steam.     The  result  is  usually 
expressed  in  pounds  of  dry  steam  evaporated  per  pound  of  coal  from  and  at 
212°  F.,  briefly  called  the  equivalent  evaporation. 

Let  the  factor  of  evaporation  be  F.  If  W  pounds  of  water  are  fed  to 
the  boiler  per  pound  of  coal  burned,  the  equivalent  evaporation  is  FW.  If 
C  be  the  heating  value  per  pound  of  fuel,  the  efficiency  is  970  FW-+-  O. 
Many  excessively  high  values  for  efficiency  have  been  reported  in  conse- 
quence of  not  correcting  for  wetness  of  the  steam ;  the  proportion  of  wet- 
ness may  range  up  to  4  per  cent,  in  overloaded  boilers.  The  highest  well- 
confirmed  figures  give  boiler  efficiencies  of  about  83  per  cent.  The  average 
efficiency,  considering  all  plants,  probably  ranges  from  0.40  to  0.60. 

573.  Checks  on  Operation.     A  careful  boiler  trial  is  rather   expensive,  and 
must  often  interfere  with  the  operation  of  the  plant.     The  best  indication  of  cur- 


CHIMNEY   DESIGN  367 

rent  efficiency  obtainable  is  that  afforded  by  analysis  of  the  flue  gases.  It  has 
been  shown  that  maximum  efficiency  is  attained  when  the  percentage  of  carbon 
dioxide  reaches  a  maximum.  Automatic  instruments  are  in  use  for  continuously 
determining  and  recording  the  proportion  of  this  constituent  present  in  flue  gases. 

574.  Boiler  and  Furnace  Efficiency.  This  measurement  (Art.  573)  in  reality 
indicates  principally  the  furnace  efficiency,  which  may  be  defined  as  the  quotient 
of  the  available  heat  (above  the  temperature  of  the  steam)  in  the  gases,  per  pound 
of  fuel  supplied,  by  the  heat  in  a  pound  of  fuel.  The  boiler  surface  efficiency,  sepa- 
rately considered,  is  then  the  quotient  of  the  heat  taken  up  by  the  steam,  by  the 
heat  originally  available  in  the  gases.  It  can  be  estimated  by  noting  the  tempera- 
ture of  the  escaping  flue  gases.  In  trials,  it  is  rarely  possible  to  accurately  distin- 
guish between  the  two  efficiencies. 

575.  Chimney  Draft.  In  most  cases,  the  high  temperature  of  the  flue  gases 
leaving  the  boiler  is  utilized  to  produce  a  natural  upward  draft  for  the  mainte- 
nance of  combustion.  At  equal  temperatures,  the  chimney  gas  would  be  heavier 
than  the  external  air  in  the  ratio  (w  +  1)-=-  n,  where  n  is  the  number  of  pounds  of 
air  supplied  per  pound  of  fuel.  If  T,  t  denote  the  respective  absolute  tempera- 
tures, then,  the  density  of  the  outside  air  being  1,  that  of  the  chimney  gas  is 

At  60°  F.,  the  volume  of  a  pound  of  air  is  13  cu.  ft.     The  weight  of 


n 
gas  in  a  chimney  of  cross-sectional  area  A  and  height  //  is  then 


The  "  pressure  head,"  or  draft,  due  to  the  difference  in  weight  inside  and  outside 
is,  per  unit  area, 


This  is  in  pounds  per  square  foot,  if  appropriate  units  are  used  ;  drafts  are,  how- 
ever, usually  stated  in  "  inches  of  water,"  one  of  which  is  equal  to  5.2  Ib.  per  square 
foot.  The  force  of  draft  therefore  depends  directly  on  the  height  of  the  chimney  ; 
and  since  n  +  1  is  substantially  equal  to  n,  maximum  draft  is  obtained  when  T-^t 
is  a  minimum,  or  (since  T  is  fixed)  when  t  is  a  maximum;  in  the  actual  case, 
however,  the  quantity  of  gas  passing  would  be  seriously  reduced  if  the  value  of  t 
were  too  high,  and  best  results  (3),  so  far  as  draft  is  concerned,  are  obtained  when 
t  :  T  :  :  25  :  12. 

To  determine  the  area  of  chimney:  the  velocity  of  the  gases  is,  in  feet  per 
second, 

v  =  V^gh  =  8.03  Vh  =  8.03\!' 

h  being  the  head  corresponding  to  the  pressure  p  and  density  d  of  the  gases  in  the 
chimney.  Also 

rf  =  -lL(i±I). 

13A    n    I 


368  APPLIED  THERMODYNAMICS 

Then  if  C  lb.  of  coal  are  to  be  burned  per  hour,  the  weight  of  gases  per  second  is 

C0  +  1>  ,  their  volume  is    C(>  +  1>, 
3600  3600  d 

and  the  area  of  the  chimney,  in  square  feet,  is 


3600  d 

A  slight  increase  may  be  made  to  allow  for  decrease  of  velocity  at  the  sides.  Most 
chimney  tables  are  based  on  an  air  supply  of  about  75  lb.  per  pound  of  fuel 
(Art.  565). 

576.  Mechanical  Draft.  In  lieu  of  a  chimney,  steam-jet  blowers  or  fans  may 
be  employed.  These  usually  cost  less  initially,  and  more  in  maintenance.  The 
ordinary  steam-jet  blower  is  wasteful,  but  the  draft  is  independent  of  weather  con- 
ditions, and  may  be  greatly  augmented  in  case  of  overload.  The  velocity  of  the 

air  moved  by  a  fan  is 

v  =  \/2  gh, 

where  h  is  the  head  due  to  the  velocity,  equal  to  the  pressure  divided  by  the 
density.  Then  ,—  v 

v  =\2  gj-  and_p  =  ^-. 

If  a  be  the  area  over  which  the  discharge  pressure  p  is  maintained,  the  work 
necessary  is  W  =  pav  =  dav*  -*•  2  g. 

We  may  note,  then,  that  the  velocity  of  the  air  and  the  amount  delivered 
vary  as  the  peripheral  speed  of  the  wheel,  its  pressure  as  the  square,  and  the 
power  consumed  as  the  cube,  of  that  speed.  Low  peripheral  speeds  are 
therefore  economical.  They  are  usually  fixed  by  the  pressure  required, 
the  fan  width  being  then  made  suitable  to  deliver  the  required  volume. 

577.  Forms  of  Fan  Draft.     The  air  may  be  blown  into  a  closed  fire  room  or 
ash  pit  or  the  flue  gases  may  be  sucked  out  by  an  induced  draft  fan.     In  the  latter 
case,  the  high  temperature  of  the  gases  reduces  the  capacity  of  the  fan  by  about 
one  half  ;  i.e.  only  one  half  the  weight  of  gas  will  be  discharged  that  would  be 
delivered  at  60°  F.     Since  the  density  is  inversely  proportional  to  the  absolute 
temperature,  the  required  pressure  can  then  be  maintained  only  at  a  considerable 
increase  in  peripheral  speed  ;  which  is  not,  however,  accompanied  by  a  concordant 
increase  in  power  consumption.     Induced  draft  requires  the  handling  of  a  greater 
weight,  as  well  as  of  a  greater  volume  of  gas,  than  forced  draft  ;  the  necessary 
pressure  is  somewhat  greater,  on  account  of  the  frictional  resistance  of  the  flues 
and  passages;  high  temperatures  lead  to  mechanical  difficulties  with  the  fans. 
The  difficulty  of  regulating  forced  draft  has  nevertheless  led  to  a  considerable 
application  of  the  induced  system. 

578.  Stokers.     Mechanical  stokers  are  often  used  when  soft  coal  is  employed 
as  fuel.     Besides  saving  some  labor,  in  large  plants  at  least,  they  give  more  per- 
fect combustion  of  hydrocarbons,  with  reduced  smoke  production.     Figure  277 


SUPERHEATERS 


369 


shows,  incidentally,  a  modern  form  of  the  old  "  Dutch  oven "  principle  for  soft 
coal  firing.  The  flames  are  kept  hot,  because  they  do  not  strike  the  relatively  cold 
boiler  surface  until  combustion  is  complete.  Fuel  is  fed  alternately  to  the  two 
sides  of  the  grate,  ro  that  the  smoking  gases  from  one  side  meet  the  hot  flame 
from  the  other  at  the  hot  baffling  "  wing  walls  "  a,  b.  The  principle  involved  in 


FIG.  277.    Arts.  578,  579.  —  Sectional  Elevation  of  Foster  Superheater  combined  with  Boiler 
and  Kent  Wing  Wall  Furnace.     (Power  Specialty  Company.) 


FIG.  278.    Arts.  578,  579.  — Babcock  and  Wilcox  Boiler  with  Chain  Grate  Stoker  and 

Superheater. 


370 


APPLIED  THERMODYNAMICS 


the  attempt  to  abate  smoke  is  that  of  all  mechanical  stokers,  which  may  be  grouped 
into  three  general  types.  In  the  chain  grate,  coal  is  carried  forward  continuously 
on  a  moving  chain,  the  ashes  being  dropped  at  the  back  end.  The  gases  from 
the  fresh  fuel  pass  over  the  hotter  coke  fire  on  the  back  portion  of  the  grate.  (See 
Fig.  278.)  The  second  type  comprises  the  inclined  grate  stokers.  The  high  com- 
bustion chamber  above  the  lower  end  of  the  grate  is  a  decided  advantage  with 
many  types  of  boilers.  The  smoke  is  distilled  off  at  the  "coking  plate."  The 
underfeed  stoker  feeds  the  coal  by  means  of  a  worm  to  the  under  side  of  the  fire, 
and  the  smoke  passes  through  the  incandescent  fuel.  All  stokers  have  the  ad- 
vantage of  making  firing  continuous,  avoiding  the  chilling  effect  of  an  open  fire 
door. 

579.  Superheaters  ;  Types.  Superheating  was  proposed  at  an  early  date,  and 
given  a  decided  impetus  by  Him.  After  1870,  as  higher  steam  pressures  were 
introduced,  superheating  was  partially  abandoned.  Lately,  it  has  been  reintro- 


FlG.  279.     Art.  579.  —  Cole  Superheater.     (American  Locomotive  Company.) 

duced,  and  the  use  of  superheat  is  now  standard  practice  in  France  and  Germany, 
while  being  quite  widely  approved  in  this  country.  Superheaters  may  be  sepa- 
rately fired,  steam  from  a  boiler  being  passed  through  an  entirely  separate  machine, 
or,  as  is  more  common,  steam  may  be  carried  away  from  the  water  to  some  space 
provided  for  it  within  the  boiler  setting  or  flue,  and  there  heated  by  the  partially 
spent  gases.  When  it  is  merely  desired  to  dry  the  steam,  the  "  superheater  "  may 
be  located  in  the  flue,  using  waste  heat  only.  When  any  considerable  increase 
of  temperature  is  desired,  the  superheater  should  be  placed  in  a  zone  of  the 
furnace  where  the  temperature  is  not  less  than  1000°  F.  With  a  difference  in 


FEED   WATER   HEATERS 


371 


mean  temperature  between  gases  and  steam  of  400°  F.,  about  5  B.  t.  u.  "may  be 

transmitted  per  degree  of  mean  temperature  difference  per  square  foot  of  surface 

per  hour  (4).     The  location  of  the  Babcock  and  Wilcox  superheater  is  shown  in 

Fig.    277  ;     a    similar    arrangement,    in 

which   the   chain    grate    stoker    is    inci- 

dentally  represented,  is  shown    in   Fig. 

278.     In  locomotive  service,  Field  tubes 

may  be   employed,   as  in  Fig.  279,   the 

steam   emerging   from   the   boiler  at  a, 

and  passing  through  the  header  b  to  the 

small  tubes  c,  c,  c,  in  the  fire  tubes  rf,  rf, 

rf(5). 

A  typical  superheater  tube  or  "ele- 
ment" is  shown  in  Fig.  280.  This  is 
made  double,  the  steam  passing  through 
the  annular  space.  Increased  heating 
surface  is  afforded  by  the  cast  iron  rings 
a,  a.  In  some  single-tube  elements,  the 
heating  surface  is  augmented  by  internal 
longitudinal  ribs.  The  tubes  should  be  located  so  that  the  wettest  steam  will 
meet  the  hottest  gases. 

580.  Feed-water  Heaters.  In  Fig.  233,  the  condensed  water  is  returned 
directly  from  the  hot  well  24,  by  way  of  the  feed  pump  IV,  to  the  boiler.  This 
water  is  seldom  higher  in  temperature  than  125°  F.  A  considerable  saving  may 
be  effected  by  using  exhaust  steam  to  further  heat  the  water  before  it  is  delivered 
to  the  boiler.  The  device  for  accomplishing  this  is  called  the  feed-water  heater. 
With  a  condensing  engine,  as  shown,  the  water  supply  may  be  drawn  from  the 
hot  well  and  the  necessary  exhaust  steam  supplied  by  the  auxiliary  exhausts  27 
and  31  ;  1  Ib.  of  steam  at  atmospheric  pressure  should  heat  about  9.7  Ib.  of 
water  through  100°.  Accurately,  W(xL  -  ?0)=  w(Q  -  q),  in  which  W  is  the 
weight  of  steam  condensed,  x  is  its  dryness,  L  its  latent  heat,  and  w  is  the  weight 
of  feed  water,  the  initial  and  final  heat  contents  of  which  are  respectively  q  and  Q. 
The  heat  contents  of  the  steam  after  condensation  are  <?0.  Then 

,  W(xL  -  y0) 


FIG.  280.    Art.  579.  —  Superheater  Element. 
(Power  Specialty  Company.) 


Q-q 

With  non-condensing  engines,  the  exhaust  steam   from  the  engines  themselves 
is  used  to  heat  the  cold  incoming  water. 

581.  Types.  Feed-water  heaters  may  be  either  "open,"  the  steam  and  water 
mixing,  or  "  closed,"  the  heat  being  transmitted  through  the  surface  of  straight 
or  curved  tubes,  through  which  the  water  circulates.  Figure  281  shows  a  closed 
heater;  steam  enters  at  A  and  emerges  at  B\  water  enters  at  C9  passes  through 
the  tubes  and  out  at  D.  The  openings  E,  E  are  for  drawing  off  condensed  steam. 
An  open  heater  is  shown  in  Fig.  282.  Water  enters  through  the  automatically 
controlled  valve  a,  steam  enters  at  b.  The  water  drips  over  the  trays,  becoming 
finely  divided  and  effectively  heated  by  the  steam.  At  c  there  is  provided  storage 


372 


APPLIED  THERMODYNAMICS 


space  for  the  mixture,  and  at  of  is  a  bed  of  coke  or  other  absorbent  material, 
through  which  the  water  filters  upward,  passing  out  at  e.  The  open  heater  usu- 
ally makes  the  water  rather  hotter,  and  lends  itself  more  readily  to  the  reclaiming 


FIG.  281.    Art.  581.  —  Wheeler  Feed  Water  Heater. 


of  hot  drips  from  the  steam  pipes,  returns  from  heating  systems,  etc.,  than  a 
heater  of  the  closed  type.  Live  steam  is  sometimes  used  for  feed-water  heating ; 
the  greater  effectiveness  of  the  boiler  heating  surface  claimed  to  arise  from  intro- 
ducing the  water  at  high  temperature  has  been  disputed  (6)  ;  but  the  high  tem- 
peratures possible  with  live  steam  are  of  decided 
value  in  removing  dissolved  solids,  and  the  waste  of 
steam  may  be  only  slight.  Closed  heaters  are; 
of  course,  used  for  this  service,  as  also  with  the 
isodiabatic  multiple-expansion  cycle  described  in 
Art.  550.  Removal  of  some  of  the  suspended  and 
dissolved  solids  is  also  possible  in  ordinary  open 
exhaust-steam  heaters.  Various  forms  of  feed- 
water  niters  are  used,  with  or  without  heaters. 


582.  The  Economizer.  This  is  a  feed-water 
heater  in  which  the  heating  medium  is  the  waste 
gas  discharged  from  the  boiler  furnace.  It  may 
increase  the  feed  temperature  to  300°  F.  or  more, 
whereas  no  ordinary  exhaust-steam  heater  can  pro- 
duce a  temperature  higher  than  212°  F.  The  gain 
by  heating  feed  water  is  about  1  B.  t.  u.  per  pound 
of  steam  for  each  degree  heated;  or  since  average 


FIG.  282.  Art.  581.  —  Open  Feed 
Heater. 

(Harrison  Safety  Boiler  Works.) 


steam  contains  1000  B.  t.  u.  net,  it  is  about  1  per  cent  for  each  10°  that  the  tem- 
perature is  raised;  precisely,  the  gain  is  (H  -  h)  •*-  Q,  in  which  Q  is  the  total  heat 
of  the  steam  gained  from  the  temperature  of  feed  to  the  state  at  evaporation  and 
h  and  H  the  total  heats  in  the  water  before  and  after  heating.  If  77,  t  be  the 
temperatures  of  flue  gases  and  steam,  respectively,  W  the  weight,  and  K  the  mean 
specific  heat  of  the  gases  (say  about  0.24),  then  the  maximum  saving  that  can  be 
effected  by  a  perfect  economizer  is  WK(T  -  t).  Good  operation  decreases  W  and 
T  and  thus  makes  the  possible  saving  small.  A  typical  economizer  installation 


THE   ECONOMIZER 


373 


is  shown  in  Fig.  283 ;  arrangement  is  always  made  for  by-passing  the  gases,  as 
shown,  to  permit  of  inspecting  and  cleaning.  The  device  consists  of  vertical  cast- 
iron  tubes  with  connecting  headers  at  the  ends,  the  tubes  being  sometimes  stag- 
gered so  that  the  gases  will  im- 
pinge against  them.  The  external 
surface  of  the  tubes  is  kept  clean 
by  scrapers,  operated  from  a  small 
steam  engine.  The  tubes  obstruct 
the  draft,  and  some  form  of  me- 
chanical draft  is  employed  in  con- 
junction with  economizers.  About 
4.8  sq.  ft.  of  economizer  surface 
are  ordinarily  used  per  boiler  horse 
power. 

583.  Miscellaneous  Devices. 
A  steam  separator  is  usually  placed 
on  the  steam  pipe  near  the  engine. 
This  catches  and  more  or  less 
thoroughly  removes  any  condensed 
steam,  which  might  otherwise 
cause  damage  to  the  cylinder. 
Steam  meters  are  being  introduced 
for  approximately  indicating  the 
amount  of  steam  flowing  through 
a  pipe.  Some  of  them  record  their 
indications  on  a  chart.  Feed-water 
measuring  tanks  are  sometimes  in- 
stalled, where  periodical  boiler 
trials  are  a  part  of  the  regular 
routine.  The  steam  loop  is  a  de- 
vice for  returning  condensed  steam 
direct  to  the  boiler.  The  drips  are 
piped  up  to  a  convenient  height, 
and  the  down  pipe  then  forms  a 
radiating  coil,  in  which  a  consid- 
erable amount  of  condensation  oc- 
curs. The  weight  of  this  column 
of  water  in  the  down  pipe  offsets  a 
corresponding  difference  in  pres- 
sure, and  permits  the  return  of 
drips  to  the  boiler  even  when  their 
pressure  is  less  than  the  boiler 
pressure.  The  ordinary  steam  trap 
merely  removes  condensed  water 
without  permitting  the  discharge  of  uncondensed  steam.  Oil  separators  are  some- 
times used  on  exhaust  pipes  to  keep  back  any  traces  of  cylinder  oil. 


374 


APPLIED   THERMODYNAMICS 


584.  Condensers.  The  theoretical  gain  by  running  condensing  is  shown  by 
the  Carnot  formula  (T  —  £)  -4-  T.  The  gain  in  practice  may  be  indicated  on  the 
P  V  diagram,  as  in  Fig.  284.  The  shaded  area  represents 
work  gained  due  to  condensation ;  it  may  amount  to  10 
or  12  Ib.  of  mean  effective  pressure,  which  means  about 
a  25  per  cent  gain,  in  the  case  of  an  ordinary  simple 
engine.  This  gain  is  principally  the  result  of  the  in- 
troduction of  cooling  water,  which  usually  costs  merely 
the  power  to  pump  it;  in  most  cases,  some  additional 
power  is  needed  to  drive  an  air  pump  as  well. 
V  In  the  surface  condenser  the  steam  and  the  water  do 

FIG  284     Art  584 Sav-  T1°^  come  m^°  contact,  so  that  impure  water  maybe  used, 

ing  Due  to  Condensation,    as  at  sea,  even  when  the  condensed  steam  is  returned  to 
the    boilers.      The    amount  of    condensing    surface  is 

usually  computed  from  Whitham's  empirical  formula  (7)  S  =  WL  -4-  180(7*  —  t), 
in  which  S  is  in  square  feet,  W  is  the  weight  of  steam  condensed  per  hour,  L  the 
latent  heat  at  the  temperature  T  of  the  steam,  arid  t  is  the  mean  temperature  of 
the  circulating  water  between  inlet  and  outlet.  Let  u,  U  be  the 
initial  and  final  temperatures  of  the  water ;  then  the  weight  w  of 
water  required  per  hour  is  WL  +(U  —  w).  The  weight  of  water 
is  often  permitted  to  be  about  40  times  the  weight  of  steam,  a 
considerable  excess  being  desirable.  The  outlet  temperature  of  the 
water  in  ordinary  surface  condensers  may  be  from  15°  to  40°  below 
that  of  the  steam. 

The  jet  condenser  is  shown  in  Fig.  285.     The  steam  and  water 
mix  in  a  chamber  above  the  air  pump  cylinder,  and  this  cylinder  is 
utilized  to  draw  in  the  water,  if  the  lift  is  not  excessive.     Here, 
theoretically,  U  =  T;  the  supply  of  water  necessary  is  less  than  in 
surface  condensers.      The  boilers  may  be  fed  from  the  hot  well, 
as  in  Fig.  233  (which  shows  a  jet 
condenser  installation),  only  when 
the  condensing  water  is  pure. 

The  siphon  condenser  is  shown 
in  Fig.  286.  Condensation  occurs 
in  the  nozzle  a,  and  the  fall  of 
water  through  b  produces  the 
vacuum.  To  preserve  this,  the 
lower  end  of  the  discharge  pipe 
must  be  sealed  as  shown.  The 
vacuum  would  draw  water  up  the 
pipe  I  and  permit  it  to  flow  over  into  the  engine,  if  it  were  not  that  the  length  cd 
is  made  31  ft.  or  more,  thus  giving  a  height  to  which  the  atmospheric  pressure 
cannot  force  the  water.  Excellent  results  have  been  obtained  with  these  con- 
densers without  vacuum  pumps.  In  some  cases,  however,  a  "  dry  "  vacuum  pump 
is  used  to  remove  air  and  vapor  from  above  the  nozzle.  The  vacuum  will  lift 
the  inlet  water  about  20  ft.  so  that,  unless  the  suction  head  is  greater  than  this,  no 
water  supply  pump  is  required,  after  the  condenser  is  started. 


FIG.  285. 


Art.  584.  —  Horizontal  Independent  Jet 
Condenser. 


CONDENSERS 


375 


585.  Evaporative  Condensers.  Steam  has  occasionally  been  condensed  by 
allowing  it  to  pass  through  coils  over  which  fine  streams  of  water  trickled.  The 
evaporation  of  the  water  (which 
may  be  hastened  by  a  fan)  cools 
the  coils  and  condenses  the  steam, 
which  is  drawn  off  by  an  air 
pump.  With  ordinary  condensers 
and  a  limited  water  supply  cooling 
towers  are  sometimes  used.  These 
may  be  identical  in  construction 
with  the  evaporative  condensers, 
excepting  that  warm  water  enters 
the  coils  instead  of  steam,  to  be 
cooled  and  used  over  again;  or 
they  may  consist  of  open  wood 
mats  over  which  the  water  falls 
as  in  the  open  type  of  feed-water 
heater.  Evaporation  of  a  portion 
of  the  water  in  question  (which 
need  not  be  a  large  proportion  of 
the  whole)  and  warming  of  the 
air  then  cools  the  remainder  of 
the  water,  the  cooling  being  facili- 
tated by  placing  the  mats  in  a 
cylindrical  tower  through  which 


FIG.  286.    Art.  o84.  —  Bulkley  Injector  Coudeuser. 
there  is  a  rapid  upward  current  of  air,  naturally  or  artificially  produced   (8). 


586.  Boiler  Feed  Pump.     This  may  be  either  steam-driven  or  power-driven 
(as  may  also  be  the  condenser  pumps).     Steam-driven  pumps,  should  be  of  the 
duplex  type,  with  plungers  packed  from  the  outside,  and  with  individually  acces- 
sible valves.     If  they  are  to  pump  hot  water,  special  materials  must  be  used  for 
exposed  parts.     The  power  pump  has  usually  three  single-acting  water  cylinders. 
There  is  much  discussion  at  the  present  time  as  to  the  comparative  economy  of 
steam-  and  power-driven  auxiliaries.     The  steam  engine  portion  of  an  ordinary 
small  pump  is  extremely  inefficient,  while  power-driven  pumps  can  be  operated,  at 
little  loss,  from  the  main  engines.     The  general  use  of  exhaust  steam  from  aux- 
iliaries for  feed-water  heating  ceases  to  be  an  argument  in  their  favor  when  econo- 
mizers are  used  ;  and  in  large  plants  the  difference  in  cost  of  attendance  in  favor 
of  motor-driven  auxiliaries  is  a  serious  item. 

587.  The  Injector.     The  pump  is  the  standard  device  for  feeding  stationary 
boilers;  the  injector,  invented  by  Giffard  about  1858,  is  used  chiefly  as  an  auxil- 
iary, although  still  in  general  application  as  the  prime  feeder  on  locomotives.     It- 
consists  essentially  of  a  steam  nozzle,  a  combining  chamber,  and  a  delivery  tube. 
In  Fig.  287,  steam  enters  at  A  and  expands  through  B,  the  amount  of  expansion 
being  regulated  by  the  valve  C.     The  water  enters  at  Z>,  and  condenses  the 
steam  in  E.     We  have  here  a  rapid  adiabatic  expansion,  as  in  the  turbine  ;  the 


376 


APPLIED  THERMODYNAMICS 


velocity  of  the  water  is  augmented  by  the  impact  of  the  steam,  and  is  in  turn  con- 
verted into  pressure  at  F.  In  starting  the  injector,  the  water  is  allowed  to  flow 
away  through  G ;  as  soon  as  the  velocity  is  sufficient,  this  overflow  closes.  An  in- 
jector of  this  form  will  lift  the  water  from  a  reasonably  low  suction  level ;  when 
the  water  flows  to  the  device  by  gravity,  the  valve  C  may  be  omitted. 


FIG.  287.    Art.  587.  —  Injector. 

A  self-starting  injector  is  one  in  which  the  adjustment  of  the  overflow  at  G  is 
automatic.  The  ejector  is  a  similar  device  by  which  the  lifting  of  water  from 
a  well  or  pit  against  a  moderate  delivery  head  (or  none)  is  accomplished.  The 
siphon  condenser  (Art.  584)  involves  an  application  of  the  injector  principle.  The 
double  injector  is  a  series  of  successive  injectors,  one  discharging  into  another. 

588.  Theory.  Tet  x,  L,  h  be  the  state  of  the  steam,  H  the  heat  in  the 
water,  and  v  its  velocity  ;  Q  the  heat  in  the  discharged  water  at  its  veloc- 
ity V.  The  heat  in  one  pound  of  steam  is  xL  +  h;  the  heat  in  one  pound 
of  water  supplied  is  IT,  and  its  kinetic  energy  v2-s-2g-y  the  heat  in  one 
pound  of  discharge  is  Q,  and  its  kinetic  energy  F2  -s-  2  g.  Let  each  pound 
of  steam  draw  in  y  pounds  of  water  ;  then 


xL  +  ft  +  y   ff  +          = 


The  values  of  ~  and  --  may  ordinarily  be  neglected,  and 


Q-H 


THE  INJECTOR  377 

In  another  form,  y(Q—II)=xL  +  h—Q,  or  the  heat  gained  by  the  water 
equals  that  lost  by  the  steam.  This,  while  not  rigidly  correct,  on  account 
of  the  changes  in  kinetic  energy,  is  still  so  nearly  true  that  the  thermal 
efficiency  of  the  injector  may  be  regarded  as  100  per  cent ;  from  this  stand- 
point, it  is  merely  a  live-steam  feed-water  heater. 

589.  Application.     The  formula  given  shows  at  once  the  relation  between 
steam  state,  water  temperature,  and  quantity  of  water  per  pound  of  steam.     As 
the  water  becomes  initially  hotter,  less  steam  is  required;  but  injectors  do  not 
handle  hot  water  well.     Exhaust  steam  may  be  used  in  an  injector :  the  pressure 
of  discharge  is  determined  by  the  velocity  induced,  and  not  at  all  by  the  initial 
pressure  of  the  steam ;  a  large  steam  nozzle  is  required,  and  the  exhaust  injector 
will  not  ordinarily  lift  its  own  water  supply. 

590.  Efficiency.     Let  S  be  the  head  against  which  discharge  is  made ; 
then  the  work  done  per  pound  of  steam   is  (1  +  y)  S  foot-pounds ;  the 
efficiency  is  S(l  -f-.y)-H  (xL  -f  h—  Q),  ordinarily  less  than  one  per  cent. 

This  is  of  small  consequence,  as  practically  all  of  the  heat  not  changed  to 
work  is  returned  to  the  boiler.  Let  W  be  the  velocity  of  the  steam  issuing  from 
the  nozzle;  then,  since  the  momentum  of  a  system  of  elastic  bodies  remains  con- 
stant during  impact,  W  +  yo  =  (1  +  y}  V.  The  value  of  W  may  be  expressed  in 
terms  of  the  heat  quantities  by  combining  this  equation  with  that  in  Art.  588.  The 
other  velocities  are  so  related  to  each  other  as  to  give  orifices  of  reasonable  size. 
The  practical  proportioning  of  injectors  has  been  treated  by  Kneass  (9). 

(1)  Finlay,  Proc.  A.  I.  E.  E.,  1907.  (2)  Trans  A.  S.  M.  E.,  XXI,  34.  (3)  Ran- 
kine,  The  Steam  Engine,  1897,  289.  (4)  Longridge.  Proc.  Inst.  M.  E.,  1896,  175. 
(5)  Trans.  A.  S.  M:  E.,  XXVIII,  10,  1606.  (6)  Bilbrough,  Power,  May  12,  1908, 
p.  729.  (7)  Trans.  A.  S.  M.  E.,  IX,  431.  (8)  Bibbins,  Trans.  A.  S.  M.E.,  XXI,  11. 
(9)  Practice  and  Theory  of  the  Injector. 

SYNOPSIS   OF   CHAPTER  XVI 

Hard  coal  requires  high  draft :  soft  coal,  a  high  rate  of  air  supply. 

In  spite  of  its  higher  cost,  commercial  factors  sometimes  make  soft  coal  the  cheaper 
fuel. 

Heating  values  :  fuel  oil,  18,000;  wood,  6500-8500;  coals,  8800-15,000;  B.  t.  u.  per  Ib. 

The  proportion  of  carbon  dioxide  in  the  flue  gases  reaches  a  maximum  when  the  air 
supply  is  just  right ;  this  is  also  the  condition  of  maximum  temperature  and  theo- 
retical efficiency. 

Advance  in  steam  power  economy  is  a  matter  of  regulation  of  air  supply ;  economy 
may  be  indicated  by  automatic  records  of  carbon  dioxide. 

Types  of  boiler :  water-tube,  horizontal  tubular,  locomotive,  marine :  conditions  of 
efficiency. 

Attention  should  be  given  to  the  circulation  of  the  gases  and  the  water. 

A  boiler  hp.  =  34|  Ib.  of  water  per  hour  from  and  at  212°  F. :  approximately  33,000 
B.  t.  u.  per  hour. 


378  APPLIED  THERMODYNAMICS 


Limit  of  efficiency  =  ;  say  0.94  ;  never  reached  in  practice. 


Boiler  efficiency  =         tfe    ;  USUally  °'4°  tO  °*6°  ' 

Furnace  efficiency  -££•£..     Heating  surface  efficiency  = 

Chimney  draft  =  H{l-^  (^)  j  .  13;  area  = 

Fan  draft  :  v-  \iTgh,  p  =  —  ,  W=  —  —  ;  slow  speeds  advantageous. 
20  20 

In  induced  draft,  the  fan  is  between  the  furnace  and  the  chimney  ;  in  forced  draft,  it 

delivers  air  to  the  ash  pit. 
Mechanical  stokers  (inclined  grate,  chain  grate,  underfeed),  used  with  soft  coal,  aim 

to  give  space  for  the  hydrocarbonaceous  flame  without  permitting  it  to  impinge  on 

cold  surfaces. 
Superheaters  may  be  located  in  the  flue,  or,  if  much  superheating  is  required,  may  be 

separately  fired.     About  5  B.  t.  u.  may  be  the  transmission  rate. 


Feed-water  heaters  may  be  open  or  closed:  w  =    -^  —  ""        ;  for  open  heaters,  q0  —  Q. 

V      * 
The   economizer  uses  the   waste  heat  of  the  flue  gases  :   saving  per  pound  of  fuel 


Condensers    may    be    surface,   jet,     evaporative,    or    siphon  :     w  =  WL  -f-  (  U  —  iC)  ; 

S  =  WL  -T-  180(  T—  t).    The   siphon  condenser  may  operate  without  a  vacuum 

pump. 

The  use  of  steam-driven  auxiliaries  affords  exhaust  steam  for  feed-water  heating. 
The    injector   converts    heat    energy    into  velocity  :   y  =  xL  +  h  ~  Q  ;    efficiency  = 

~ 


xL  +  h  —  Q 


PROBLEMS 


1.  One  pound  of  pure  carbon  is  burned  in  16  Ib.  of  air.     Assuming  reactions  to  be 
perfect,  find  the  percentage  composition  of  the  flue  gases  and  the  rise  in  temperature, 
the  specific  heats  being,  CO2,  0.215  ;  N,  0.245  ;  O,  0.217. 

2.  A  boiler  evaporates  3000  Ib.  of  water  per  hour  from  a  feed-water  temperature 
of  200°  F.  to  dry  steam  at  160  Ib.  pressure.     What  is  its  horse  power  ? 

3.  In  Problem  2,  what  proportion  of  the  whole  heat  in  the  fuel  is  carried  away 
in  the  flue  gases,  if  their  temperature  is  600°  F.,  assuming  the  specific  heats  of  the 
gases  to  be  constant  ?    The  initial  temperature  of  the  fuel  and  air  supplied  is  0°  F. 

4.  The  boiler  in  Problem  2  burns  350  Ib.  of  coal  (14,000  B.  t.  u.  per  pound)  per  hour. 
What  is  its  efficiency  ? 

5.  In  Problems  1  and  3,  the  temperature  of  the  steam  is  350°  F.     Find  the  furnace 
efficiency  and  the  efficiency  of  the  heating  surface  (Art.  574). 

6.  In  Problem  1,  if  the  gas  temperature  is  600°  F.,  the  air  temperature  60°  F., 
compare  the  densities  of  the  gases  and  the  external  air.     What  must  be  the  height  of  a 
chimney  to  produce,  under  these  conditions,  a  draft  of  1  in.  of  water?     Find  the 
diameter  of  the  (round)  chimney  to  burn  5000  Ib.  of  coal  per  hour. 


THE   STEAM   POWER  PLANT  379 

7.  Two  fans  are  offered  for  providing  draft  in  a  power  plant,  15,000  cu.  ft.  of 
air  being  required  at  1^  oz.  pressure  per  minute.     The  first  fan  has  a  wheel  30  in.  in 
diameter,  exerts  1  oz.  pressure  at  740  r.  p.  m.,  delivers  405  cu.  ft.  per  minute,  and  con- 
sumes 0.16  hp.,  both  per  inch  of  wheel  width  and  at  the  given  speed.     The  second  fan 
has  a  54-inch  wheel,  runs  at  410  r.  p.  m.  when  exerting  1  oz.  pressure,  and  delivers 
726  cu.  ft.  per  minute  with  0.29  hp.,  both  per  inch  of  wheel  width  and  at  the  given 
speed.     Compare  the  widths,  speeds,  peripheral  speeds,  and  power  consumptions  of  the 
two  fans  under  the  required  conditions. 

8.  Dry  steam  at  350°  F.  is  superheated  to  450°  F.  at  135  Ib.  pressure.     The  flue 
gases  cool  from  900°  F.  to  700°  F.    Find  the  amount  of  superheating  surface  to  provide 
for  3000  Ib.  of  steam  per  hour,  and  the  weight  of  gas  passing  the  superheater.    If  280 
Ib.  of  coal  are  burned  per  hour,  what  is  the  air  supply  per  pound  of  coal  ? 

9.  Water  is  to  be  raised  from  60°  F.  to  200°  F.  in  a  feed-water  heater,  the  weight 
of  water  being  10,000  Ib.  per  hour.     Heat  is  supplied  by  steam  at  atmospheric  pressure, 
0.95  dry.     Find  the  weight  of  strain  condensed  (a)  in  an  open  heater,  (6)  in  a  closed 
heater.    Find  the  surface  necessary  in  the  latter  (Art.  584). 

10.  In  Problem  3,  what  would  be  the  percentage  of  saving  due  to  an  economizer 
which  reduced  the  gas  temperature  to  400°  F.  ? 

11.  An  engine  discharges  10,000  Ib.  per  hour  of  steam  at  2  Ib.  absolute  pressure, 
0.95  dry.     Water  is  available  at  60°  F.    Find  the  amount  of  water  supplied  for  a  jet 
condenser.    Find  the  amount  of  surface,  and  the  water  supply,  for  a  surface  condenser 
in  which  the  outlet  temperature  of  the  water  is  85°  F.     If  the  surface  condenser  is 
operated  with  a  cooling  tower,  what  weight  of  water  will  theoretically  be  evaporated  in 
the  tower,  assuming  the  entire  cooling  to  be  due  to  such  evaporation.     (N.B.    A  large 
part  of  the  cooling  is  in  practice  effected  by  the  air.) 

12.  Find  the  dimensions  of  the  cylinders  of  a  triplex  single-acting  feed  pump  to 
deliver  100,000  Ib.  of  water  per  hour  at  60°  F.  at  a  piston  speed  of  100  ft.  per  minute 
and  30  r.  p.  m. 

13.  Dry  steam  at  100  Ib.  pressure  delivers  3000  Ib.  of  water  per  hour  from  an  injec- 
tor at  165°  F.,  the  inlet  temperature  of  the  water  being  60°  F.    Find  the  weight  of 
steam  used.     The  water  is  measured  on  the  inlet  side  of  the  injector. 

14.  In  Problem  13,  the  boiler  pressure  is  100  Ib.     What  is  the  efficiency  of  the 
injector,  considered  as  a  pump  ? 

15.  In  Problem  13,  the  velocity  of  the  entering  water  is  12  ft.  per  second,  that  of  the 
discharge  is  1 14  ft.  per  second.    Find  the  velocity  of  the  steam  leaving  the  discharge 
nozzle. 

16.  What  is  the  relation  of  altitude  to  chimney  draft  ?     (See  Problem  12,  Chapter 
XIII.) 


CHAPTER   XVII 

DISTILLATION  — FUSION- LIQUEFACTION  OF  GASES 
VACUUM  DISTILLATION 

591.  The  Still.     Figure  288  represents  an  ordinary  still,  as  used  for 
purifying  liquids  or  for  the  recovery  of  solids  in  solution  by  concentration. 
Externally  applied  heat  evaporates  the  liquid  in  A,  which  is  condensed  at 

»-  B.      All   of    the    heat   ab- 

13 

sorbed  in  A  is  given  up  at 

B  to  the  cooling  water; 
the  only  wastes,  in  theory, 
arise  from  radiation.  Con- 
ceive the  valve  c  to  be 
closed,  and  the  space  from 
the  liquid  level  d  to  this 
valve  to  be  filled  with  satu- 
rated vapor,  no  air  being 
present  in  any  part  of  the 
apparatus.  Then  when  the 
value  c  is  opened,  a  vacuum  will  gradually  be  formed  throughout  the 
system,  and  evaporation  will  proceed  at  lower  and  lower  temperatures. 

Since  the  total  heat  of  saturated  vapor  decreases  with  decrease  of 
pressure,  evaporation  will  thus  be  facilitated.  In  practice,  however,  the 
apparatus  cannot  be  kept  free  from  air;  and,  notwithstanding  the  opera- 
tion of  the  condenser,  the  vacuum  would  soon  be  lost,  the  pressure  increas- 
ing above  that  of  the  atmosphere.  This  condition  is  avoided  by  the  use 
of  a  vacuum  pump,  which  may  be  applied  at  e,  removing  air  only ;  or,  in 
usual  practice,  at/,  removing  the  condensed  liquid  as  well.  Evaporation 
now  proceeds  continuously  at  low  pressure  and  temperature.  The  possi- 
bility of  utilizing  low-temperature  heat  now  leads  to  marked  economy. 

592.  Application.     Vacuum  distillation  is  employed  on  an  important  scale  in 
sugar  refineries,  soda  process  paper-pulp  mills,  glue  works,  glucose  factories,  for 
the  preparation  of  pure  water,  and  in  the  manufacture  of  gelatine,  malt  extract, 

380 


FIG.  288.    Art.  591.  —  Still. 


DISTILLATION 


381 


382 


APPLIED  THERMODYNAMICS 


wood  extracts,  caustic  soda,  alum,  tannin,  garbage  products,  glycerine,  sugar  of 
milk,  pepsin,  and  licorice.  In  most  cases,  the  multiple-effect  apparatus  is  employed 
(Art.  594). 

593.  Newhall  Evaporator.     This  is  shown  in  Fig.  289.     Steam  is  used 
to  supply  heat ;  it  enters  at  A,  and  passes  through  the  chambers  A1,  A2, 
to  the  tubes  B,  B.     After  passing  through  the  tubes,  it  collects  in  the 
chambers  (72,  (71,  from  which  it  is  drawn  off  by  the  trap  D.     The  liquid 
to  be  distilled  surrounds  the  tubes.     The  vapor  forms  in  E,  passes  around 
the  baffle  plate  F  and  out  at  G.    The  concentrated  liquid  is  drawn  off  from 
the  bottom  of  the  machine. 

594.  Multiple-effect  Evaporation.     Conceive  a  second  apparatus 
to  be  set  alongside  that  just  described ;    but  instead  of  supplying 


mr 


FIG.  290.    Art.  595.  —  Triple  Effect  Evaporator. 


steam  at  A,  let  the  vapor  emerging  from  G-  of  the  first  stage  be 
piped  to  A  in  the  second,  and  let  the  liquid  drawn  off  from  the  bot- 


MULTIPLE-EFFECT   EVAPORATION 


383 


torn  of  the  first  be  led  into  the  second  ;  then  further  evaporation  may 
proceed  without  the  expenditure  of  additional  heat,  the  liquid  being 
partially  evaporated  and  the  vapor  partially  condensed  by  the  inter- 
change of  heat  in  the  second  stage,  the  pressure  in  the  shell  (outside 
the  tubes)  being  less  than  that  in  the  first  stage.  The  construction  will 
be  more  clearly  understood  by  reference  to  Fig.  290. 

595.  Yaryan  Apparatus.     Here  the  heat  is  applied  outside  the  tubes, 
the  liquid  to  be  distilled  being  inside.     The  liquid  is  forced  by  a  pump 
through  a  small  orifice 

at  the  end  of  the  tube, 
breaking  into  a  fine 
spray  during  its  pas- 
sage. The  fine  sub- 
division and  rapid 
movement  of  the 
liquid  facilitate 
the  transfer  of  heat. 
The  baffle  plates  E, 
E,  Fig.  291,  serve  to 
separate  the  1  i  quid  and  FIG.  291.  Art.  595.  —  Yaryan  Evaporator, 

its  vapor,  the  former 

settling  in  the  chamber  6,  the  latter  passing  out  at  c.  Figure  290  shows  a 
"  triple-effect "  or  three-stage  evaporator ;  steam  (preferably  exhaust 
steam)  enters  the  shell  of  the  first  stage.  The  liquor  to  be  evaporated 
enters  the  tubes  of  this  stage,  becomes  partly  vaporized,  and  the  separated 
vapor  and  liquid  pass  off  as  just  described.  From  the  outlet  c,  Fig.  291, 
the  vapors  pass  through  an  ordinary  separator,  which  removes  any  ad- 
ditional entrained  liquid,  discharging  it  back  to  b,  aud  then  proceed  to 
the  shell  of  the  second  stage.  Meanwhile  the  liquid  from  the  chamber  b 
of  the  first  stage  has  been  pumped,  through  a  hydrostatic  tube  which 
permits  of  a  difference  in  pressure  in  two  successive  sets  of  tubes,  into  the 
tubes  of  the  second  stage.  As  many  as  six  successive  stages  may  be  used ; 
the  vapors  from  the  last  being  drawn  off  by  a  condenser  and  vacuum 
pump.  The  liquid  from  the  chamber  b  of  the  last  stage  is  at  maximum 
density. 

596.  Condition   of   Operation.     The   vapor   condensed   in   the   various 
shells  is  ordinarily  water,  which  in  concentrating  operations  may  be  drawn 
off  and  wasted,  or,  if  the  temperature  is  sufficiently  high,  employed  in  the 
power  plant.     The  condenser  is  in  communication  with  the  last  tubes,  and, 
through  them,  with  all  of  the  shells  and  tubes  excepting  the  first  shell  j  but 


384  APPLIED   THERMODYNAMICS 

between  the  various  stages  we  have  the  heads  of  liquid  in  the  chambers  6, 
which  permit  of  carrying  different  pressures  in  the  different  stages.  A 
gradually  decreasing  pressure  and  temperature  are  employed,  from  first  to 
last  stage;  it  is  this  which  permits  of  the  further  boiling  of  a  liquid 
already  partly  evaporated  in  a  former  effect.  The  pressure  in  the  tubes  of 
any  stage  is  always  less  than  that  in  the  surrounding  shell ;  the  pressure 
in  the  shell  of  any  stage  is  equal  to  that  in  the  tubes  of  the  previous  stage. 

597.  Theory.     Let  IF  be  the  weight  of  dry  steam  supplied;  the 
heat  which  it  gives  up  is  WL.     Let  w  be  the  weight  of  liquid  enter- 
ing the  first  stage,  H  its  heat,  and  h  and  I  the  heat  of  the  liquid  and 
latent  heat  corresponding  to  the  pressure  in  the  first-stage  tubes.     If 
x  pounds  of  this  liquid  are  evaporated  in  the  first  stage,  the  heat 
supplied  is  xl  +  wQi  —  H),  theoretically  equal  to  WL;  whence 

x=  \WL-wQi-  IT)]  -^  I. 

Then  x  pounds  of  vapor  enter  the  shell  of  the  second  stage,  giving 
up  the  heat  xl.  The  weight  of  liquid  entering  the  tubes  of  the 
second  stage  is  w  —  x.  Let  the  latent  heat  and  heat  of  liquid  at 
the  pressure  in  the  tubes  of  this  stage  be  m  and  i:  then  the  heat  ab- 
sorbed, if  y  pounds  be  evaporated,  is  ym  -j-  (w  —  x)(i  —  A),  the  last 
term  being  negative,  since  i  is  less  than  h.  Then 

y  =  [xl  —  (w  —  x)(i  —  A)]  -r-  m. 

Consider  now  a  third  stage.  The  heat  supplied  may  be  taken  at  ym  ; 
the  heat  utilized  at 

zM  +  (w  -  x  -  y)(I-  0, 

(z  being  the  weight  of  liquid  evaporated,  M  its  latent  heat,  and  J  the 
corresponding  heat  of  the  liquid), 

whence  z  =  [ym  —  (w  —  x  —  y)(I  —  z)]  -*-  M. 

The  analysis  may  be  extended  to  any  number  of  stages. 

598.  Rate  of  Evaporation.     Qrdinarily,  the  evaporated  liquid  is  an   aqueous 
solution;  the  total  evaporation  per  pound  of  steam  supplied  increases  with  the 
number  of  stages,  being  practically  limited  by  the  additional  constructive  expense 
and  radiation  loss.     For  a  triple-effect  evaporator,  the  total  evaporation  per   W 
pounds  of  steam  supplied  is  x  +  y  +  z.     Let  W  =  1,  and  let  the  steam  be  supplied 
at  atmospheric  pressure,  the  vacuum  at  the  condenser  being  0.1  Ib.  absolute,  and 
the  successive  shell  pressures  14.7,  8.1,  1.5.     The  pressures  in  the  tubes  are  then 
8.1,  1.5,  and  0.1 :  whence  L=  970.4,  1  =  987.9,  A  =  151. 3,  m  =  1027.8,  i  =81.9,  7=6.98, 


GOSS  EVAPORATOR  385 

M  =  1048.1.  Let  H  be  100,  the  liquid  being  supplied  at  132°  F.  A  definite  re- 
lation must  exist  between  w  and  W,  in  order  that  the  supply  of  vapoi;  to  the  last 
effect,  y,  may  be  sufficient  to  produce  evaporation,  yet  not  so  great  as  to  burden  the 
apparatus;  this  is  to  be  determined  by  the  degree  of  concentration  desired  in  any 
particular  case,  whence  a:  +  y  +  z  =  (f)w,  in  which  (/)  represents  the  proportion  of 
liquid  to  be  evaporated.  Let  (/)  =  1.0,  as  is  practically  the  case  in  the  distillation 
of  water;  then  w  =  x+y  +  z.  We  now  have,  x  =  0.982 -0.0521  tc,  y  =  0.88  +  0.0211  w, 
z  =  0.726  +  0.094 w,  x  +y  +  z  =  w  =  2.588  -I-  0.063 w,  whence  w  =  2.76.  This  is 
equivalent  to  about  27.6  Ib.  of  water  evaporated  per  pound  of  coal  burned  under 
the  best  conditions.  By  increasing  the  number  of  effects,  evaporation  rates  up  to 
37  Ib.  have  been  attained  in  the  triple-effect  machine. 

599.  Efficiency.      The    heat    expended     in     evaporation     is     in    this    case 
a;/-|-yw  +  2lf  =3080  B.  t.  u.     The  heat  supplied  by  the  steam  was  WL  =970.4  B.  t.  u. 
The  efficiency  is,  therefore,  apparently  3.18,  a  result  exceeding  unity.     A   large 
amount  of  additional  heat  has,  however,  been  furnished  by  the  substance  itself,  which 
is  delivered,  not  as  a  vapor,  but  as  a  liquid,  at  the  condenser. 

600.  Water  Supply.     The  condenser  being  supplied  per  pound  of  steam 
supplied  to  the  first  stage  with  v  pounds  of  water,  its  heat  increasing  from 
?i  to  Nj  the  heat  interchange  is  z3f=v(N—n),  whence,  v=zM+  (N—n),  the 
liquid  being  discharged  at  the  boiling  point  corresponding  to  the  pressure 
in  the  condenser.     In  this  case,  for  N—n  =  25,  v  =  40.2  Ib.,  or  the  water 
supply  is  40.2  -f-  2.76  =  14.5  Ib.  per  pound  of  liquid  evaporated.     Some  ex- 
cess is  allowed  in  practice :  the  greater  the  number  of  effects,  the  less,  gen- 
erally speaking,  is  the  quantity  of  water  required. 

601.  The  Goss  Evaporator.     This  is  shown  in  Fig.  292.     Steam  enters 
the  first  stage  F  from  the  boiler  (7,  say  at  194  Ib.  pressure  and  379°  F. 
The  liquid  to  be  evaporated  (water)  here  enters  the  last  stage  A,  say  at 
62°  F. ;  the  boiling  of  the  liquid  in  each  successive  stage  from  F  to  A 
produces  steam  which  passes  to  the  interior  tube  of  the  next  succeeding 
stage,  along  with  the  water  resulting  from  condensation  in  the  interior  tube 
of  the  previous  stage.     The  condensed  steam  from  the  first  stage,  is,  how- 
ever, returned  to  the  boiler,  which  thus  operates  like  a  house-heating  boiler, 
with  closed  circulation.     Let  1  Ib.  of  liquid  be  evaporated  in  F;  its  pressure 
and  temperature  are  so  adjusted  that,  in  this  case,  the  whole  temperature 
range  between  that  of  the  steam  (379°  F.)  and  that  of  the  liquid  finally  dis- 
charged from  A  (213°  F.)  is  equally  divided  between  the  stages.     The 
amount  of  vapor  produced  in  any  stage  may  then  be  computed  from  the 
heat  supplied  for  the  assigned  temperature  and  corresponding  pressure. 
Finally,  in  A,  no  evaporation  occurs,  the  incoming  liquid  being  merely 
heated ;  and  it  is  found  that  the  weights  of  discharged  liquid  and  incoming 
liquid  are  equal,  amounting  each  to  4.011  Ib.     The  steam  supplied  by  the 


386 


APPLIED  THERMODYNAMICS 


FUSION  387 

boiler  may  be  computed  ;  in  F,  we  condense  steam  at  379°  F.,  at  which  its 
latent  heat  per  pound  is  845.8.  It  is  assumed  that  3  per  cent  of  the  heat 
supplied  in  each  effect  is  lost  by  evaporation;  the  available  heat  in  each  pound 
of  steam  supplied  is  then  0.97  x  845.8  =  820.426.  This  heat  is  expended  in 
evaporating  1  Ib.  of  water  at  312.6°  to  dry  steam  at  345.8°,  requiring 


1187.44  -  282.26  =  905.18  B.  t.  u.,  for  which  =  i.i  Ib.  of  steam  are 

820.43 

required.      The  whole  evaporation  for  the  six-effect  apparatus  is  —  -  = 

3.646  Ib.  per  pound  of  steam.  For  the  second  effect,  E,  the  heat  supplied 
is  £345.8  =  870.66,  gross,  or  0.97  x  870.66  =  844.54,  net.  The  heat  utilized 
is  1.873(282.22  -  248.7)  +  (0.873  x  895.18)  =  844.54.  In  D,  the  heat  supplied 
is  0.97  [(0.873  X  Z312.6)  +  1(316.98  -282.22)]  =790.8;  that  utilized  is 
2.633(248.7  -215.3)  +  (0.76  x  918.42)  =  790.8.  The  heat  interchange  is 
perfect  ;  it  should  be  noted  that  the  liquid  to  be  evaporated  and  the  heat- 
ing medium  are  moving  in  opposite  directions.  This  involves  the  use  of  a 
greater  amount  of  heating  surface,  but  leads  to  higher  efficiency,  than  the 
customary  arrangement.  An  estimated  economy  of  60  Ib.  of  water  per 
pound  of  coal  is  possible  with  seven  stages  (1). 

FUSION 
602.    Change  of  Volume  during  Change  of  State.     The  formula 


y     v  = 

T    dP 

was  derived  in  Art.  368.  The  specific  volume  of  a  vapor  below  the  criti- 
cal temperature  exceeds  that  of  the  liquid  from  which  it  is  produced; 

dT 

consequently  V—  v  has  in  all  cases  a  positive  value,  and  hence  —  must 

be  positive;  i.e.  increase  of  pressure  causes  an  increase  in  temperature. 
It  is  universally  true  that  the  boiling  points  of  substances  are  increased  by 
increase  of  pressure,  and  vice  versa,  at  points  below  the  critical  tempera- 
ture. If  for  any  vapor  we  know  a  series  of  corresponding  values  of  V,  L, 
T,  and  v,  we  may  at  once  find  the  rate  of  variation  of  temperature  with 
pressure. 

603.  Fusion.  The  same  expression  holds  for  the  change  of  state  de- 
scribed as  fusion  ;  the  Carnot  cycle,  Figs.  162,  163,  may  represent  melting 
along  abj  adiabatic  expansion  of  the  liquid  along  be,  solidification  along 
cdj  and  adiabatic  compression  of  the  solid  to  its  melting  point  along  da. 
In  this  case,  V  does  not  always  exceed  v  ;  it  does  for  the  majority  of  sub- 
stances, like  wax,  spermaceti,  sulphur,  steariiie,  and  paraffin,  which  con- 
tract in  freezing  ;  and  for  these,  we  may  expect  to  find  the  melting  point 


388 


APPLIED  THERMODYNAMICS 


raised  by  the  application  of  pressure.  This  has,  in  fact,  been  found  to  be 
the  case  in  the  experiments  of  Bunsen  and  Hopkins  (2).  On  the  other 
hand,  those  few  substances,  like  ice,  east  iron,  and  bismuth,  which  expand 
in  freezing,  should  have  their  melting  points  lowered  by  pressure  ;  a  result 
experimentally  obtained,  for  ice,  by  Kelvin  (3)  and  Mousson  (4).  The 
melting  point  of  ice  is  lowered  about  0.0135°  F,  for  each  atmosphere  of 
pressure.  The  expansion  of  ice  in  freezing  is  of  practical  consequence.  A 
familiar  illustration  is  afforded  by  the  bursting  of  water  pipes  in  winter. 

604.    Comments.    As  the  result  of  a  number  of  experiments  with  non-metallic 
substances,  Person  (5)  found  the  following  empirical  formula  to  hold  : 


in  which  L  is  the  latent  heat  of  fusion,  C,  c  are  the  specific  heats  in  the 
liquid  and  solid  states  respectively,  and  T  the  Fahrenheit  temperature  of  fusion. 
Another  general  formula  is  given  for  metals.  A  body  may  be  reduced  from  the 
solid  to  the  liquid  state  by  solution.  This  operation  is  equivalent  to  that  of  fusion, 
but  may  occur  over  a  wider  range  of  temperatures,  and  is  accompanied  by  the  ab- 
sorption of  a  different  quantity  of  heat.  The  applications  of  the  fundamental 
formulas  of  thermodynamics  to  the  phenomena  of  solution  have  been  shown  by 
Kirchoff  (6).  The  temperature  of  fusion  is  that  highest  temperature  at  which  the 
substance  can  exist  in  the  solid  state,  under  normal  pressure.  The  latent  heat  of 
fusion  of  ice  has  a  phenomenally  high  value. 

LIQUEFACTION  OF  GASES 

605.    Graphical  Representation.     In  Fig.  293,  let  a  represent  the 
state  of  a  superheated  vapor.     It  may  be  reduced  to  saturation,  and 
liquefied,  either  at  constant  pressure,  along  acd, 
the  temperature  being  reduced,  or  at  constant 
temperature  along  abe,  the  pressure  being  in- 
creased.     After  reaching  the  state  of  satura- 
tion,   any   diminution    of   volume    at   constant 
temperature,  or  any  de- 
L     crease  in  temperature  at 

FiG.293.  Art.605.— Lique-  constant  volume,    must 

faction  of    Superheated 

Vapor.  produce  partial  lique- 

faction. Constant  tem- 
perature liquefaction  is  not  applicable  to  gases 
having  low  critical  temperatures.  Thus,  in 
Fig.  294,  ab  is  the  liquid  line  and  cd  the  FIG.  294.  Art.  oos.-Lique- 

„  faction    and  Critical  Tem- 

saturation  curve  of  carbon  dioxide,  the  two      perature. 


LIQUEFACTION   OF   GASES  389 

meeting  at  the  critical  temperature  of  88°  F.  From  the  state  e 
this  substance  can  be  liquefied  only  by  a  reduction  in  temperature. 
With  "permanent"  gases,  having  critical  temperatures  as  low  as 
—  200°  C.,  an  extreme  reduction  of  temperature  must  be  effected 
before  pressure  can  cause  liquefaction. 

606.  Early  Experiments.     Monge  and  Clouet,- prior  to  1800,  had  liquefied  sul- 
phur dioxide,  and  Northmore,  in  1805,  produced  liquid  chlorine  and  possibly  also 
sulphurous  acid,  in  the  same  manner  as  was  adopted  by  Faraday,  about  1823,  in 
liquefying  chlorine,  hydrogen  sulphide,  carbon  dioxide,  nitrous  oxide,  cyanogen, 
ammonia,  and  hydrochloric  acid  gas.     The  apparatus  consisted  simply  of  a  closed 
tube,  one  end  of  which  was  heated,  while  the  other  was  plunged  in  a  freezing  mix- 
ture.    Pressures  as  high  as  50  atmospheres  were  reached.     Colladon  supplemented 
this  apparatus  with  an  expansion  cock,  the  sudden  fall  of  pressure  through  the 
cock  cooling  the  gas;  and  in  Cailletet's  hands  this  apparatus  led  to  useful  results. 
Thilorier,  utilizing  the  cooling  produced  by  the  evaporation  of  liquid  carbon  diox- 
ide, first  produced  that  substance  in  the  solid  form.     Natterer  compressed  oxygen 
to  4000  atmospheres,  making  its  density  greater  than  that  of  the  liquid,  but  with- 
out liquefying  it.     Faraday  obtained  minimum  temperatures  of  —  166°  F.  by  the 
use  of  solid  carbon  dioxide  and  ether  in  vacuo. 

607.  Liquefaction  by  Cooling.     Andrews,  in  1849,  recognizing  the 
limiting  critical  temperature,  proposed  to  liquefy  the  more  permanent 
gases  by  combining  pressure  and  cooling.      Figure  295  shows  the 
principle  involved.       Let   the   gas   be   com- 
pressed isothermally  from  P  to  «,  expanded 

through  an  orifice  along  a6,  re-compressed  to 
c,  again  expanded  to  rf,  etc.  A  single  cycle 
might  suffice  with  carbon  dioxide,  while 
many  successive  compressions  and  expansions 


would  be  needed  with  a  more  permanent  gas.  FIG.  295.    Art.  GOT.  — 
The  process  continues,  in  all  cases,  until  the     £^ng  by   Pressure   and 
temperature  falls   below  the   critical  point ; 

and  at  x  the  substance  begins  to  liquefy.  The  action  depends  upon 
the  cooling  resulting  from  unrestricted  expansion.  With  an  abso- 
lutely perfect  gas,  no  cooling  would  occur ;  the  lines  ab,  cd,  etc., 
would  be  horizontal,  and  this  method  of  liquefaction  could  not  be 
applied.  The  "  perfect  gas,"  in  point  of  fact,  could  not  be  liquefied. 
All  common  gases  have  been  liquefied. 

608.   Modern  Apparatus.     Cailletet  and  Pictet,  independently,  in  1877, 
succeeded  in   liquefying  oxygen.      The   Pictet   apparatus   is    shown   in 


390 


APPLIED   THERMODYNAMICS 


Fig.  296.     The  jacket  a  was  filled  with  liquid  sulphur  dioxide,  from  which 

the  vapor  was  drawn  off  by  a  pump,  and  delivered  to  the  condenser  b. 

The  compressor  c  re-delivered  this 
substance  in  the  liquid  condition 
to  the  jacket,  cooling  in  d  a  quan- 
tity of  carbon  dioxide  which  was 
itself  compressed  in  e  and  used  as 
a  cooling  jacket  for  the  oxygen 
gas  in  /.  The  oxygen  was  formed 
in  the  bomb  g,  and  expanded 
through  the  cock  7i,  producing  a 
fall  of  temperature  which,  sup- 

FiG.296.  Art.  608,  Prob, 7. -Cascade  System,  plemented  by  the  cooling  effect 

of  the  carbon  dioxide,  produced 

liquid  oxygen.      The  series  of  cooling  agents  used,  suggested  the  name 

cascade  system. 

609.  Dewar's  Experiments.  Dewar  liquefied  air  in  1884  and  nitrogen  about 
1892.  In  1895  he  solidified  air  by  free  expansion,  producing  a  jellylike  substance. 
In  1896  he  obtained  liquid  hydrogen,  by  the  use  of  which  air  and  oxygen  were 
solidified,  forming  white  masses.  A  temperature  of  —  396.4°  F.  was  obtained. 
Dewar's  final  apparatus  was  that  of  Pictet,  but  compressors  were  used  to  deliver 
the  gases  to  the  liquefying  chamber,  and  ethylene  was  employed  in  place  of  car- 
bon dioxide. 


610.  Regenerative  Process  ;  Liquid 
Air.  The  fall  of  temperature  ac- 
companying a  reduction  in  pressure 
has  been  utilized  by  Linde  (7)  and 
others  in  the  manufacture  of  liquid 
air.  In  the  first  form  of  apparatus, 
shown  in  Fig.  297,  air  was  com- 
pressed to  about  2000  Ib.  pressure  in 
a  three-stage  machine  J.,  and  after 
cooling  in  B  was  delivered  to  the 
inner  tube  of  a  double  coil  (7,  through 
which  it  passed  to  the  expansion 
valve  D.  Here  a  considerable  fall 
of  temperature  took  place.  The 
cooled  and  expanded  air  then  passed 
back  through  the  outer  tube  of  the 


FIG.  297. 


Art.  610.  —  Liquefaction   of 
Air. 


LIQUID  AIR  391 

coil,  cooling  the  air  descending  the  inner  tube,  and  was  discharged 
at  F.  The  effect  was  cumulative,  and  after  a  time  liquid  air  was 
deposited  in  E.  In  the  present  type  of  machine,  the  compressor 
takes  its  supply  from  jP,  a  decided  improvement.  The  regenerative 
principle  has  been  adopted  in  the  recent  forms  of  apparatus  of 
Hampson,  Solvay,  Dewar,  and  Tripler. 

The  latent  heat  of  evaporation  of  air  at  atmospheric  pressure  is  about  140 
B.  t.  u.  (8).  In  its  commercial  form,  it  contains  small  particles  of  solid  carbon 
dioxide;  when  these  are  removed  by  nitration,  the  liquid  becomes  clear.  The 
boiling  point  of  nitrogen  is  somewhat  higher  than  that  of  oxygen;  fairly  pure 
liquid  oxygen  may,  therefore,  be  obtained  by  allowing  liquid  air  to  partially 
evaporate  (9).  The  cost  of  production  of  liquid  air  has  been  carefully  estimated 
in  one  instance  to  approach  22  cents  per  pint  (10). 

(1)  Trans.  A.  S.  M.  E.,  XXV,  03.  The  steam  table  used  was  Peabody's,  1890  ed. 
The  temperatures  noted  on  Fig.  292  are  approximate :  those  in  the  text  are  correct. 
(2)  Sep.  B.  A.,  1854,  II,  56.  (3)  Phil.  Mag.,  1850:  III,  xxxvii,  123.  (4)  Des- 
chanel,  Xatural  Philosophy  (Everett  tr.),  1893,  II,  331.  (5)  Ann.  de  Chem.  et  de 
Phys.,  November,  1849.  (6)  Pogg.  Ann.,  1858.  (7)  Zeuner,  Technical  Thermody- 
namics (Klein),  II,  303-313;  Trans.  A.  S.  M.  E.,  XXI,  156.  (8)  Jacobus  and  Dick- 
erson :  Trans.  A.  8.  M.  E.,  XXI,  166.  (9)  See  the  very  complete  paper  by  Rice, 
Trans.  A.  S.  M.  E.,  XXI,  156.  (10)  Tests  of  a  Liquid  Air  Plant,  Hudson  and  Gar- 
land ;  University  of  Illinois  Bulletin,  V,  16. 

SYNOPSIS   OF  CHAPTER   XVII 

Distillation 

The  still  is  a  device  for  purifying  liquids  or  recovering  solids  by  partial  evaporation. 
By  evaporation  in  vacuo,  the  heat  consumed  may  be  reduced  in  many  important 

applications :  •  waste  heat  may  be  employed. 
Steam  may  supply  the  heat ;  in  the  Newhall  apparatus,  the  steam  circulates  through 

tubes. 

In  the  Yaryan  apparatus,  the  steam  surrounds  the  tubes. 
The  vapors  rising  from  the  solution  may  supply  the  heat  required  in  a  second  "  effect," 

provided  that  the  solution  there  is  under  a  less  pressure  than  in  the  first  stage. 
As  many  as  six  stages  are  used,  the  pressure  on  the  solution  decreasing  step  by  step. 
Evaporation  per  effect:  *  =  "1  -  .(»- J)  .  ,,«!-(»-»)«-»), 

„  _ym-(w  —  x-  y) ( J—  Q 
M 

In  a  typical  case,  the  triple-effect  machine  gives  an  evaporation  of  2.76  Ib.  per  pound  of 
steam. 

Water  required  at  the  condenser  per  pound  of  liquid  evaporated  — — 

In  the  Goss  evaporator,  the  steam  and  the  solution  move  in  opposite  directions  ;  this 
increases  the  necessary  amount  of  surface,  but  also  the  efficiency. 


392  APPLIED  THERMODYNAMICS 

Fusion 

7  7  ft    7"    /7  7* 

The  formula  V—  v  =     °         -    applies  to  fusion.    The  melting  points  of  substances 

may  be  either  raised  or  lowered  by  the  application  of  pressure,  according  as  the 
specific  volume  in  the  liquid  state  is  greater  or  Zess  than  that  in  the  solid  state. 

The  melting  point  of  ice  is  lowered  about  0.0135°  F.  per  atmosphere  of  pressure  imposed. 

L=(C—  c)(!T+  256)  for  non-metallic  substances. 

Liquefaction  of  Gases 

A  vapor  below  the  critical  temperature  may  be  liquefied  either  at  constant  pressure  or 

at  constant  temperature. 

No  substance  can  be  liquefied  unless  below  the  critical  temperature. 
A  few  common  substances  have  been  liquefied  by  the  use  of  pressure  and  freezing 

mixtures. 

A  further  lowering  of  temperature  is  produced  by/ree  expansion. 
Liquefaction  may  be  accomplished  with  actual  gases  by  successive  compressions  and 

free  expansions. 
The  Pictet  apparatus  (cascade  system)  employed  the  latent  heat  of  vaporization  of 

successive  fluids  to  cool  more  volatile  fluids. 
The  regenerative  system  provides  for  the  free  expansion  of  a  highly  compressed  gas 

previously  reduced  to  atmospheric  temperature.     This  is  used  in  manufacturing 

liquid  air. 

PROBLEMS 

1.  Water  entering  a  still  at  40°   F.   is  evaporated,  (a)  at  atmospheric   pressure, 
(6)  at  2  Ib.  absolute  pressure.     What  is  the  saving  in  heat  in  the  latter  case  ?     What 
more  important  saving  is  possible  ? 

2.  Water  entering  a  double-effect  evaporator  at  80°  F.  is  completely  distilled,  the 
steam  supplied  being  dry  and  at  atmospheric  pressure,  the  pressure  in  the  second-stage 
shell  being  8  Ib.  and  that  in  the  second-stage  tubes  1  Ib.     Cooling  water  is  available  at 
60°  F.    The   temperature  of  the  circulating  water  at  the   condenser    outlet  is  80°. 
Find  the  steam  consumption  per  pound  of  water  evaporated  and  the  cooling  water 
consumption,  if  the  vacuum  pump  discharge  is  at  85°  F. 

3.  In  Fig.  292,  take  temperatures  as  given  ;  assume  one  pound  of  water  to  be  com- 
pletely evaporated  in  F,  and  complete  condensation  to  occur  in  the  inner  tube  of  each 
effect ;  and  compute,  allowing  3  per  cent  for  radiation,  as  in  Art.  601  : 

(a)  The  weight  of  steam  condensed  in  F. 

(6)  The  weight  of  steam  evaporated  in  JE1,  and  of  water  delivered  to  E. 
(c)  The  weight  of  boiler  steam  used  per  pound  of  water  evaporated  in  the  whole 
apparatus.    Use  the  steam  tables  on  pp.  247,  248. 

4.  The  weight  of  one  cubic  foot  of  H2O  at  32°  F.  and  atmospheric  pressure  being 
57.5  Ib.  as  ice  and  62.42  Ib.  as  water,  and  the  latent  heat  of  fusion  -of  ice  being  142 
B.  t.  u.,  find   how  much  the  melting  point  of  ice  will  be  lowered  if  the  pressure  is 
doubled  (Art.  603). 

5.  The  specific  heat  of  ice  being  0.504,  find  its  latent  heat  of  fusion  at  32°  F.  from 
Art.  604. 


PROBLEMS  393 

6.  How  much  liquid  air  at  atmospheric  pressure  would  be  evaporated  in  freezing 
1  Ib.  of  water  initially  at  60°  F.  ? 

7.  In  a  Pictet  apparatus,  Fig.  296,  1  Ib.  of  air  is  liquefied  at  atmospheric  pressure, 
free  expansion  having  previously  reduced  its  temperature  to  the  point  of  liquefaction. 
The  condensation  is  produced  by  carbon  dioxide,  which  evaporates  in  the  jacket  with- 
out change  of  temperature,  at  such  a  pressure  that  its  latent  heat  of  vaporization  is 
200  B.  t.  u.     How  many  pounds  of  carbon  dioxide  are  evaporated  ?     This  dioxide  is 
subsequently  liquefied,  at  a  higher  pressure  and  whjle  dry  (latent  heat  =  120),  and 
cooled  through  100Q  F.     Its  specific  heat  as  a  liquid  may  be  taken  as  0.4.    The  lique- 
faction and  cooling  of  the  carbon  dioxide  are  produced  by  the  evaporation  of  sulphur 
dioxide  (latent  heat  220  B.  t.  u.).     What  weight  of  sulphur  dioxide  will  be  evaporated 
per  pound  of  air  liquefied  ?     Why  would  the  operation  described  be  impracticable  ? 

8.  From  Art.  245,  find  the  fall  of  temperature  at  expansion  in  a  Linde  air  machine 
in  which  the  air  is  compressed  to  2000  Ib.  absolute  and  cooled  to  60°  F.,  and  then  ex- 
panded to  atmospheric  pressure.     How  many  complete  circuits  must  the  air  make  in 
order  that  the  temperature  may  fall  from  60°  F.  to  —  305°  F.,  if  the  same  fall  of  tem- 
perature is  attained  at  each  circuit  ? 

9.  Plot  on  the  entropy  diagram  the  path  of  ice  heated  at  constant  pressure  from 
—  400° F.  to  32°  F.,  assuming  the  specific  heat  to  be  constant,  and  then  melted  at 
atmospheric  pressure.     How  will  the  diagram  be  changed  if  melting  occurs  at  a  pres- 
sure of  1000  atmospheres  ? 

Plot  a  curve  embracing  states  of  the  completely  melted  ice  for  a  wide  range  of  pres- 
sures. Construct  lines  analogous  to  the  constant  dryness  lines  of  the  steam  entropy 
diagram  and  explain  their  significance. 

10.  At  what  temperature  will  the  latent  heat  of  fusion  of  ice  be  0  ?  What  would 
be  the  corresponding  pressure  ? 


CHAPTER   XVIII 


MECHANICAL  REFRIGERATION 

611.  History.     Refrigeration  by  " freezing  mixtures"  has  been  practiced  for 
centuries.    Patents  covering  mechanical  refrigeration  date  back  at  least  to  1835  (1). 
In  the  first  machines,  ether  was  the  working  substance,  and  the  cost  of  operation 
was  high.     Pictet  introduced  the  use  of  sulphur  dioxide  and  carbon  dioxide.    The 
transportation  of  refrigerated  meats  began  about  1873  and  developed  rapidly  after 
1880,  most  of  the  earlier  machines  using  air  as  a  working  fluid.     The  possibility 
of  safely  shipping  refrigerated  fresh  fruits,  milk,  butter,  etc.,  has  revolutionized 
the  distribution  of  these  food  products ;  and,  to  a  large  extent,  refrigerating  pro- 
cesses have  eliminated  the  use  of  ice  in  breweries,  packing  houses,  fish  and  meat 
markets,  hotels,  etc.     The  two  important  applications  of  artificial  refrigeration  at 
present  are  for  the  production  of  artificial  ice  and  for  cold  storage. 

612.  Carnot   Cycle   Reversed.      In   Fig.    298,   let   the   cycle   be 
worked  in  a  counter-clockwise  direction.     Heat  is  absorbed  along 
dc  and  emitted  along  la\  the  latter  quantity  of   heat  exceeds  the 
former  by  the  work  expended,  abed.     The   object  of  refrigeration 
is  to  cool  some  body.     This  cooling  may  be  produced  by  a  flow  of 


FIG.  298.    Art.  612.  —  Reversed  Carnot  Cycle. 

heat  from  the  body  to  the  working  fluid  along  d  c.  Cyclic  action  is 
possible  only  under  the  condition  that  the  working  fluid  afterward 
transfer  the  heat  to  some  second  body  along  la.  The  body  to  be 

394 


REGENERATIVE  REFRIGERATION  395 

cooled  is  called  the  vaporizer ;  the  second  body,  which  in  turn  re- 
ceives heat  from  the  working  fluid,  is  the  cooler.  The  heat  taken 
from  the  vaporizer  is  ndcN\  that  discharged  to  the  cooler  is  nabN. 
The  function  of  the  machine  is  to  cause  heat  to  pass  from  the  vapor- 
izer to  a  substance  warmer  than  itself;  i.e.  the  cooler.  This  is 
accomplished  without  contravention  of  the  second  law  of  thermo- 
dynamics, by  reason  of  the  expenditure  of  mechanical  work.  The 
refrigerating  machine  is  thus  a  heat  pump. 

The  Carnot  cycle,  with  a  gas  as  the  working  fluid,  would  lead  to  an  exces- 
sively bulky  machine  (Art.  249).     Early  forms  of  apparatus  therefore  embodied 
the  regenerative  principle  (Art.  257).     This 
is  illustrated  in  Fig.  299. 

Without  the  regenerator,  air  would 
be  compressed  adiabatically  from  1  to 
2,  cooled  at  constant  pressure  along 
2  3,  expanded  adiabatically  along  3  4, 
and  allowed  to  take  up  heat  from  the 
body  to  be  refrigerated  along  41.  In 
practice,  this  heat  is  partly  taken  from 

the  body,  and  partly  from  other  sur- 

,.    J'      .  .    J          *  FIG.  299.      Art.   612.  —  Regenerative 

rounding    objects    after   the   working  Refrigeration. 

air  has  left  this  body,  say  at  5.     The 

absorption  of  heat  along  51  then  effects  no  good  purpose.  If,  however, 
this  part  of  the  heat  be  absorbed  from  the  compressed  air  at  3,  that 
body  of  air  may  be  cooled,  in  consequence,  along  3  6,  so  that  adiabatic  ex- 
pansion will  reduce  the  temperature  to  that  at  7,  lower  than  that  at  4. 
This  is  accomplished  by  causing  the  air  leaving  the  cooler  to  come  into 
transmissive  contact  with  that  leaving  the  vaporizer.  The  effect  of  the 
regenerator  is  cumulative,  increasing  the  fall  of  temperature  at  each  step ; 
but  since  the  expansion  cylinder  must  be  kept  constantly  colder  as  expan- 
sion proceeds,  a  limit  soon  arises  in  practice. 

In  Kirk's  machine  (1863),  a  compressing  cylinder  was  used  for  the  operation  c&, 
Fig.  298,  and  two  expansive  cylinders  for  the  operation  ad,  one  receiving  the  air 
from  each  end  of  the  compressor  cylinder.  The  pressure  throughout  the  cycle  was 
kept  considerably  above  that  of  the  atmosphere,  and  temperatures  of  —  39°  F.  were 
obtained.  The  regenerator  consisted  of  layer?  of  wire  gauze  located  in  the  pis- 
tons (2).  The  air  machines  of  Hargreaves  and  Inglis  (1878),  Tuttle  and  Lugo, 
Lugo  and  McPherson,  Hick  Hargreaves,  Stevenson,  Haslam,  Lightfoot,  Hall,  and 
Cole  and  Allen,  have  been  described  by  Wallis-Tayler  (3).  The  Bell-Coleman  ma- 
chine may  be  regarded  as  the  forerunner  of  all  of  these,  although  many  variations 
in  construction  and  method  of  working  have  been  introduced. 


396 


APPLIED  THERMODYNAMICS 


613.  Bell-Coleman  Machine.  This  is  the  Joule  air  engine  of  Art.  101, 
reversed.  It  operates  in  the  net  cycle  given  by  an  air  compressor  and  an 
air  engine,  as  in  Art.  213.  In  Figs.  300  and  301,  C  is  the  room  to  be 
cooled,  A  a  cooler,  M  a  compressor,  and  N  an  expansive  cylinder  (air 
engine).  In  the  position  shown,  with  the  pistons  moving  toward  the  left, 
air  flows  from  C  to  M  at  the  temperature  Tc.  On  the  return  stroke,  the 
valve  a  closes,  the  air  is  compressed  along  cb,  Fig.  301,  and  the  valve  s 


FIG.  300.    Art.  613.  —  Bell-Coleman 
Machine. 


FIG.  301.     Arts.  613,  614,  616,  622,  623. 
—  Reversed  Joule  Cycle. 


opens,  permitting  of  discharge  into  A  along  be,  at  the  temperature  Tb. 
The  operation  is  now  repeated,  the  drawing  in  of  air  from  C  to  M  being 
represented  by  the  line  fc.  Meanwhile  an  equal  weight  of  air  has  been 
passing  from  A  to  N  at  the  temperature  Ta,  less  than  Tb  on  account  of  the 
action  of  the  cooler,  along  ea  ;  expanding  to  the  pressure  in  C  along  ad, 
reaching  the  temperature  Td,  lower  than  that  in  C;  and  passing  into  C  at 
constant  pressure  along  df.  The  work  expended  in  the  compressor  cylinder 
isfcbe;  that  done  by  the  expansion  cylinder  isfead;  the  difference,  abed, 
represents  work  required  from  without  to  permit  of  the  cyclic  operation. 
If  the  lines  ad,  be,  are  isodiabatics, 


Tc      Td' 

Suitable  means  are  provided  for  cooling  the  air  in  the  compressor  cylinder,  so  as  to 
avoid  the  losses  due  to  a  rise  of  temperature  (Art.  195),  and  also  for  drying  the 
air  entering  the  expansion  cylinder. 

614.  Analysis  of  Action.  Let  air  at  147  Ib.  pressure  and  60°  F., 
at  a,  Fig.  301,  expand  adiabatically  behind  a  piston  along  ad,  until 
its  pressure  is  14.7  Ib.  Its  temperature  at  d  is 

*±3 
-J]  "   =  519.6  -*-  (10)  °'2875  =  269°  absolute  or  -  191°  F. 

*  d/ 

Let  this  cold  air  absorb  heat  along  do  at  constant  pressure,  until  its 


BELL-COLEMAN   MACHINE 


397 


temperature  rises  to  0°  F.     Then  let  it  be  compressed  adiabatically 
until  its  pressure  is  again  147  lb.,  along  cb.     Since 

=  890°  absolute,  or  430°  F. 


The  air  now  rejects  heat  at  constant  pressure  along  ba  to  cold  water, 
or  some  other  suitable  agent,'  and  the  action  recommences.  In 
practice,  the  paths  ad  and  be  are  not  adiabatic,  n  <  y,  and  the  changes 
of  temperature  are  less  than  those  just  computed. 


615.  Entropy  Diagram.  Let  aenfbc,  ¥ig.  302,  represent  the  pv  and  nt 
diagrams  of  a  Bell-Coleman  machine  working  in  two  corapressive  stages. 
Choosing  the  point  c  on  the  entropy  plane  arbitrarily  as  to  entropy,  but  in 
its  proper  vertical  location,  we  plot  the  line  of  constant  pressure  ca  up  to 
the  line  of  temperature  at  a.  Then  ae  is  drawn  as  an  adiabatic,  intersected 


FIG.  302.    Art.  615.  —  Two-stage  Joule  Cycle. 

by  the  constant  pressure  curve  ne,  with  nf,  cb,  and  bf  as  the  remaining 
paths.  The  area  aenfbc  measures  the  expenditure  of  work  to  effect  the 
process.  Along  ca,  theoretically,  heat  is  taken  from  the  cold  chamber  to 
the  extent  cgha.  The  work  expended  in  single-stage  compression  would 
have  been  camb.  We  have  then  the  following  ratios  of  heat  extracted  to 
work  expended: 

cgha  t  j_ j. •_„     cqha 


single-stage  compression, 


camb 


two-stage  compression, 


aenfbc 


616.  Work  of  Compression.  In  Fig.  301,  for  M  pounds  of  air 
circulated  per  minute,  the  heat  withdrawn  from  the  cold  chamber 
along  dc  is  Q  —  Mk(Tc  —  Td}.  The  work  expended  in  compression  is 

7,^7:         ,  P»Vb-PeVe  r,\       Mn 


n-l 


n-l 


Mn  f  ^  ~^r 


M 


398  APPLIED  THERMODYNAMICS 

If   compression    is    adiabatic,    n  =  y,  f^V^'-J,  PCVC=  RTC, 


R=k      —       and  Wc  =  MkTc        -l    =  Mk(  Tb  -  Tc).     Similarly, 

\  y  J  \1c      J 

for    the    engine   (clearance  being    ignored    in    both   cases),  WE  = 
Mh(Ta  —  TO).     The  net  work  expended  is  then 


Wc-  WE= 

We  might  also  write,  heat  delivered  to  the  cooler  =q  =  Mk(Tb—  Ta), 
W-  W  =    -     = 


617.  Cooling  Water.     The  heat  carried  away  at  the  cooler  must  be  equal 
to  the  heat  extracted  along  dc  plus  the  heat  equivalent  of  the  net  work 
expended  ;  it  is 

Mk(Tc  -  Td  +  Tb  -  Tc-Ta  +  Td)  =  M  k(Tb  -  Tu), 

as  the  path  indicates.     Let  the  rise  in  temperature  of  the  cooling  water  be 
T  —  t  :  then  the  weight  of  water  required  is  Mk(Tb  —  Ta)  -f-  (T  —  t). 

618.  Size  of  Cylinders.     At  N  revolutions  per  M  pounds  of  air 
circulated,  the  displacement  per  stroke  of  the  double-acting   com- 

pressor  piston  must  be,  ignoring  clearance,  D  —  MVC  -+-2N= 

The  same   air  must  pass  through  the  expansion  cylinder  ;  its  dis- 

MRT  T 

placement  is      w*  ;   the  two  displacements  have  the  ratio  -^  if  the 

^  ""JT4  J.  d 

cylinders  run  at  equal  r.  p.  m. 

The  piston  displacements  may  be  corrected  for  clearance  as  in  Art.  233.  They 
should  be  further  increased  from  5  to  10  per  cent  to  allow  for  imperfect  valve  action, 
etc.  A  slight  drop  in  pressure  at  the  end  of  expansion  is  not  objectionable.  The 
temperature  Td  and  the  capacity  of  the  machine  may  be  varied  by  changing  the 
point  of  cut-off  of  the  expansion  cylinder. 

619.  Practical  Proportions.     In  air  machines  of  the  so-called  "  open  type,"  the 
pressure  in  the  cold  chamber  is  that  of  the  atmosphere  ;  the  temperature  may  be 
anywhere  between  0  and  50°  F.     The  maximum  pressure  is  often  made  four  at- 
mospheres absolute.     The  cooling  water  may  be  warmed  from  60  to  80°  F.,  and  the 
air  may  leave  the  condenser  at  90°  F.     Clearance  may  be  from  2  per  cent  upward  ; 
piston  speeds  range  from  75  to  300  ft.   per    minute,    according   to   the   type   of 
compressor. 

620.  Objections  to  Air  Machines.     The  size  of  apparatus  is  inordinate  as  com- 
pared with  that  of  the  vapor-  compression  machines  to  be  described.     The  size  may 


COEFFICIENT   OF   PERFORMANCE  399 

be  considerably  reduced  by  operating  under  pressure,  as  in  the  Kirk  and  Allen 
"  dense  air  "  machines,  in  which  the  suction  pressure  exceeds  that  of  the  atmosphere. 
Small  machines  of  the  latter  type  are  frequently  used  in  marine  service  for  cooling 
pantries  and  for  making  ice  for  table  use.  The  suction  pressure  is  about  65  lb., 
the  discharge  pressure  225  lb.  Coils  must  be  used  in  the  vaporizer.  The  regenera- 
tive modification  (Art.  612)  may  be  applied,  resulting  in  temperatures  as  low  as 
—  80°  F.  Much  difficulty  has  been  experienced  in  air  machines  from  the  presence 
of  water  vapor,  which  congeals  in  the  pipes  and  passages  at  low  temperatures. 
Lightfoot  (4)  has  introduced  a  form  in  which  expansion  is  conducted  in  two  stages. 
The  temperature  of  the  air  in  the  first  stage  is  reduced  to  only  about  35°  F.,  at 
which  most  of  the  vapor  is  precipitated  and  carried  off,  before  the  air  enters  the 
second  cylinder.  In  many  air  machines,  ordinary  mechanical  separators  are  used 
to  dry  the  air. 

621.  Coefficient  of  Performance.      In  all  cases,  we  have  the  relation 
heat  taken  from  the  cold  body  -f  work  done  =  heat  rejected  to  the  cooler ;  or 
Q  +  W=  q.     The  ratio  Q  H-  W  is  described  as  the  coefficient  of  performance. 
For  the  Carnot  cycle,  it  is  obviously  t  -f-  (  T  —  t),  the  limiting  values  being 
unity  and  infinity.     This  ratio  is  sometimes  spoken  of  as  the  efficiency,  a 
designation  sufficiently  correct  so  far  as  work  expenditure  goes,  but  which 
is  apparently  not  in  conformity  with  the  prin- 

ciple  that  no  physical  transformation  can  have 

an   efficiency    equalling  unity.      Figure  3026 

explains   the  anomaly.     The  Carnot  cycle  is 

abed;    an  and  bN  are    indefinite    adiabatics. 

Now    ndcN-s-  abed  =  Q  -^  W    may    have    any 

value  whatever  exceeding  1;    but   these,  two 

areas  do  not  represent  all  of  the  heat  actions 

occurring   in   the    cycle.     Heat   has   been   re- 

moved  by  the  condenser  along  60,  equivalent 

to  nabN=q.     We  may  indefinitely  lower  the 

"  efficiency  "  by  increasing  the  upper  temperature,  as  by  the  paths  ef,  gh, 

etc.,   without  at   all  increasing  the  useful  refrigerating  effect  obtained. 

TVe-  may,  in  fact,  regard  refrigeration  as  a  negative  effect  produced  by  the 

cooling   in   the   condenser,  the  negative  work  done  being   regarded  as  a 

by-product  of  this  cause :   —  q  =  —  Q  —  W.     A  reversal  of  the  argument 

of  Art.  139  serves  to  show  that  no  cycle  can  give  a  higher  coefficient  of 

performance  than  that  of  Carnot. 

622.  Desirable  Range.     The  value  of  the  coefficient  of  performance  is 
increased  as  that  of  (T—t)  decreases;  i.e.,  for  efficient  refngeration,  the 
range  of  temperature  must  be  small,  a  result  of  extreme  practical  impor- 
tance.    It  is  more  economical  to  cool  the  given  body  of  air  or  other  sub- 
stance directly  through  the  required  range  of  temperature,  than  to  cool 


400  APPLIED  THERMODYNAMICS 

one  tenth,  say,  of  this  body,  through  ten  times  the  temperature  range, 
afterward  cooling  the  remainder  by  mixture.  This  is  a  special  example 
of  the  general  therm odynamic  principle  that  mixtures  of  substances  at 
different  temperatures  are  wasteful,  such  processes  being  irreversible.  In 
practice,  T  is  fixed  by  the  temperature  of  the  cooling  water.  It  is  seldom 
less  than  60°  F.  The  refrigerant  temperature  t  should  then  be  kept  as 
high  as  possible,  for  the  service  in  question,  if  operation  is  to  be  efficient ; 
it  must,  however,  be  somewhat  below  the  desired  room  or  solution  tem- 
perature, in  order  that  the  heat  transfer  may  be  reasonably  rapid.  In 
making  ice,  for  example,  t  must  be  considerably  below  32°  F. 

A  reversal  of  the  demonstration  in  Art.  255,  as  applied  to  Fig.  301, 
shows  that  the  coefficient  of  performance  for  the  Joule  cycle  (Bell-Cole- 

T  T 

man  machine),  with  adiabatic  paths,  is  — — ^=-  =  ~ — ^-j   for  the  corre- 

Ta—  id      lb  —  *c 

spending  Carnot  cycle  it  would  have  been  Tc-t-(Ta— Tc),  a  naturally 
higher  value.* 

Since  any  heat  motor  using  air  is  bulky,  it  is  necessary,  in  order  to  keep  the 
size  of  these  machines  within  reasonable  limits,  to  make  the  temperature  range 
large.  This  lowers  the  coefficient  of  performance,  which  in  practice  is  usually 
only  about  one  fifth  that  of  a  good  ammonia  refrigerating  machine.  Air,  how- 
ever, is  the  least  expensive  of  fluids,  is  everywhere  obtainable,  is  safe,  and  may  be 
worked  at  high  temperatures  without  excessive  pressure. 

623.  The  Kelvin  Warming  Machine.  In  Fig.  301,  let  an  air  engine  receive  its 
supply  along  ea  at  normal  temperature  and  high  pressure.  The  air  expands  along  ad, 
falling  in  temperature,  after  which  it  is  warmed  by  transmission  from  the  external 
atmosphere  along  dc  and  compressed  in  a  separate  cylinder  along  cb.  The  tem- 
perature at  c  is  equal  to  that  at  a.  The  compression  along  cb  increases  the  tem- 
perature, and  the  hot  air  may  be  discharged  into  coils  in  an  apartment  to  be 
heated.  The  ratio  of  heating  done  to  power  expended  is 

Tb      Ta  Ta 

Tb  -  Ta  -  Tc+  Td      Ta  -  T* 

The  entropy  diagram  is  that  of  Fig.  302,  and  the  ratio  of  heat  delivered  to  the 
room  to  work  expended  is  here  bmhg  -4-  bmac,  which  exceeds  unity,  because  of  the 
heat  supplied  by  the  external  air.  This  is  consequently  an  ideal  method  for  heat- 
ing. Its  advantage  increases  as  the  range  of  temperature  decreases.  Considering 
an  ideal  heat  engine  and  an  ideal  warming  machine,  both  working  in  the  same 
Carnot  cycle,  the  combined  efficiency  so  far  as  power  is  concerned  would  be  unity. 
The  efficiency  would  exceed  that  of  direct  stove  heating  without  any  loss  whatever, 
whenever  the  range  of  temperature  in  the  engine  exceeded  that  in  the  warming 
machine.  Practically,  the  economical  range  of  temperature  would  be  low,  the 
machine  of  immense  size,  and  the  operation  slow. 

*  Tc  is  the  highest  temperature  at  which  refrigeration  may  be  performed  ;  and  Ta  is 
the  lowest  temperature  at  which  the  cooling  water  is  effective. 


VAPOR  REFRIGERATION 


401 


624.  The  Vapor  Compression  Machine.     In  the  air  machine,  the  temperature 
is  reduced  by  expansion  in  a  working  cylinder.     The  mere  flow  of  the  air  through 
a  valve  would  not  perceptibly  lower  its  temperature  (Art.  73).     With  a  vapor,  a 
decided  lowering  of  temperature  occurs  when  the  pressure  is  reduced  by  free 
expansion.     The  expansion  cylinder  may,  therefore,  be  omitted,  and  this  omission 
is  made  in  spite  of  the  fact  that  an  opportunity  for  saving  some  power  is  thereby 
lost. 

625.  Principle.     If  a  small  quantity  of  ether  be  poured  into  the 
palm  of  the  hand,  a  sensation  of  cold  is  produced.     This  is  due  to 
the  rapid  evaporation  of  the  ether  at  the  temperature  of  the  body ; 
the  heat  thus  absorbed  by  the  ether  is  received  from  the  hand,  de- 
creasing the  temperature  of  the  latter.     In  Fig.  303,  let  the  closed 


FIG.  303.     Art.  625.  —  Vapor  Refrigeration. 

vessel  R  be  partly  filled  with  a  liquid  at  the  temperature  £,  having 
above  it  its  saturated  vapor.  Then  the  pressure  in  R  will  be  that 
at  which  the  boiling  point  of  the  liquid  is  t.  If  the  liquid  is  anhy- 
drous ammonia,  for  example,  and  t  =  68°  F.,jt?  =  125.056  Ib.  absolute. 
Let  some  of  the  liquid  pass  through  E  to  the  condensing  coil  B,  in 
which  the  pressure  is  P,  less  than  p.  Its  heat  per  pound  tends  to 
change  from  h  to  H\  since  h  exceeds  H,  a  certain  amount  of  liquid 
must  be  evaporated  in  B  to  reestablish  thermal  equilibrium  ;  thus, 

h  =  H+  XL,  or  X  =  *—-^. 


If,  now,  the  coil  B  be  immersed  in  water  at  a  temperature  higher 
than  its  own,  the  remaining  (1  —  X)  pounds  of  liquid  may  evapo- 


402 


APPLIED  THERMODYNAMICS 


rate  ;  the  surrounding  water  will  be  cooled,  giving  up  heat  (1  —  X)L 
if  the  substance  in  the  coil  be  completely  evaporated,  and  the  pres- 
sure in  B  be  kept  constantly  at  P,  by  artificially  removing  the  added 
vapor  from  B  as  rapidly  as  it  is  formed.  The  substance  used  must 
be  one  having  a  low  boiling  point  even  under  heavy  pressure,  if  the 
surrounding  water  is  to  be  cooled  much  below  the  temperature  of 
the  air. 


FIG.  304. 


Art.    626.  — Vapor   Compression 
Machine. 


626.    Action  of  Compressor.     In  Fig.  304,  A  represents  the  com- 
pressor, B  the  condenser,  0  the  vaporizer,  and  D  the  expansion  valve. 

_ The    compressor    piston    first 

moves  upward,  drawing  in  vapor 
from  C.  On  the  return  stroke, 
the  valve  e  is  closed  (the  valves 
are,  in  practice,  built  in  the  com- 
pressor cylinder)  and  the  vapor 
is  compressed.  When  its  pres- 
sure equals  that  in  B,  the  valve 
/  is  opened,  and  discharge  oc- 
curs. The  valve /is  now  closed 

and  D  is  opened,  the  pressure  falling  from  that  in  B  to  that  in  C. 
Described  as  a  plant  cycle,  vapor  is  compressed  along  cb,  Fig.  305, 
condensed  in  the  condenser  along  ba,  becoming  liquid  at  a,  and  ex- 
pands through  the  valve  D  along  ad, 
its  pressure  falling  so  that  it  begins 
to  boil  violently.  Further  boiling 
gives  the  path  do,  along  which  heat 
is  removed  from  the  vaporizer  C. 
Refrigeration  begins  at  d,  as  soon  as 
the  vapor  has  passed  the  expansion 
valve.  The  pipes  beyond  this  valve 
are  usually  covered  with  snow.  The 
vapor  process  always  involves  (1) 
the  condensation  of  the  vapor,  (2)  a  lowering  of  its  pressure  and 
temperature  by  expansion,  (3)  evaporation  of  the  liquid  in  the 
vaporizer,  and  (4)  compression  to  the  initial  state.  The  under- 
lying principles  are  two :  the  raising  of  the  boiling  point  by  pres- 


FIG.  305.     Art.  626.  —  Vapor  Cycle. 


VAPOR  REFRIGERATION 


403 


404 


APPLIED  THERMODYNAMICS 


sure,  and  the  absorption  of  heat  from  surrounding  bodies  during 
evaporation.  The  pump  analogy  is  useful.  The  vaporizer  may  be 
likened  to  a  pit  or  well  in  which  a  fixed  water  level  is  to  be  main- 
tained ;  by  using  a  pump,  the  water  may  be  raised  to  a  level  at  which 
it  will  of  itself  flow  away.  The  "  pump  "  is  the  compressor,  which 
raises  the  low-temperature  heat  of  the  vaporizer  to  a  high-tempera- 
ture heat  which  can  flow  away  with  the  condensed  water.  The 
heat  absorbed  by  the  water  is  usually  valueless  for  further  service, 
as  its  temperature  seldom  exceeds  80°  F. 

Figure  306  represents  a  complete  plant. 
The  pipes  a,  b  correspond  to  those  similarly 
lettered  in  Fig.  304.      The  vaporizer   may 
e/\  be  merely  an  insulated  room  to  be  cooled, 

or  a  vessel  of  water  or  brine  the  temperature 
of  which  is  to  be  lowered.  There  should  be 
no  loss  of  liquid  in  operation  excepting  by 
leakage. 


V 


627.  Entropy   Diagram.       Figure 
JTO        .  307  shows  the  various  forms  of  en- 

F,o.   307.      Arts.  627,   628,    62!,,    630  -    ^W  diagram'  according  as  the  Sub- 
Vapor    Refrigeration,    Entropy    Dia-    stance      is      Wet      {dcla,     dgefa)      dry 

(dy^fa),  or  superheated  (djklfa)  as 

it  leaves  the  vaporizer.  These  are  based  on  adiabatic  paths.  The 
actual  operation  is  not  a  perfect  Clausius  cycle.  During  expansion 
the  condition  is  one  of  constant  total 
heat,  giving  such  a  path  as  axd,  Fig. 
308.  This  decreases  the  useful  re- 
frigerating effect  area  to  ydjz.  Com- 
pression may  be  made  more  economical 
than  adiabatic,  as  in  air  compressors, 
by  jacketing  or  spraying  with  oil  or 
other  liquid ;  the  compressive  path  may 
then  be,  say,  /&,  decreasing  the  work  ex- 
penditure to  axdjb,  without  altering  the 
refrigerating  effect.  The  path  jl,  if 
represented  exponentially,  will  show  a  value  of  n  less  than  that  of 
y  for  the  vapor  in  question.  An  actual  indicator  diagram  from  a 
vapor  compressor  is  given  in  Fig.  309. 


FIG.  308.    Art.  627.  —  Modifications 
of  Refrigerative  Cycle. 


VAPOR  REFRIGERATION 


405 


628.  Coefficient  of  Performance.  For  the  cycle  dcba  of  Fig.  307, 
in  which  the  vapor  at  no  state  becomes  superheated,  maximum  heat 
removed  from  the  vaporizer  is,  say, 
xl.  Heat  is  returned  to  it,  however, 
along  ad,*  the  liquid  being  lowered 
in  temperature,  to  the  extent  H—  h. 
The  net  refrigerating  effect  is 

FIG.  309.      Art.  627.  —  Ammonia  Com- 
0  =  xl  —  (H —  Jl).  pressor  Indicator  Diagram. 

The  heat  delivered  to  the  condenser  is  XL,  and  the  work  done  is 


The  coefficient  of  performance  is  then 


-=  (xl  - 
W 


+  (XL 


Formulas  may  readily  be  derived  for  the  coefficient  when  the  vapor 
becomes  superheated  during  compression  or  even  before  compression 
begins. 

629.  Multi-stage  Operation  :  Superheat.  A  gain  is  possible  by  compress- 
ing in  two  or  more  stages.  This  gives  an  entropy  diagram  like  that  of  Fig.  309  a. 
Fig.  307  shows  that  the  highest  coefficient  of  performance  is  attained  when  the 

vapor  remains  saturated  (wet  or  dry), 
throughout  the  cycle.  Comparing  the  cycles 
abed  and  afegd,  for  example,  the  added  re- 
frigeration effect  cgnm  is  gained  at  the  cost 
of  the  proportionately  greater  expenditure 
of  external  work  cgefb.  Superheating  may 
be  prevented  by  keeping  the  vapor  always 
sufficiently  wet  at  the  beginning  of  com- 
pression, or  by  cooling  during  compression 
so  as  to  avoid  the  adiabatic  path,  as  de- 
scribed in  Art.  198.  "Dry"  compression 
(in  which  superheating  occurs)  involves 
the  use  of  jackets  to  permit  of  lubrication. 
Wet  compression  is  far  more  frequently 
practiced. 


FIG.  309  a.    Art.  629.  —  Two-stage 
Compression. 


630.    Choice  of  Liquid.     The  entropy  diagram,  Fig.  307,  shows  clearly 
one  consideration  which  should  influence  the  choice  of  a  working  fluid. 


*  In  this  ideal  case,  no  cooling  occurs  between  the  condenser  and  the  expansion 
valve  or  between  the  expansion  valve  and  the  vaporizer. 


406 


APPLIED  THERMODYNAMICS 


The  net  refrigerating  effect  is  reduced  by  the  area  under  da,  as  explained 
in  Art.  627.  The  steeper  this  line,  the  less  the  reduction ;  the  longer  the 
line  dc,  the  greater  is  the  refrigerating  effect.  Steepness  of  the  line  da 
means  a  low  specific  heat  of  liquid;  a  long  line  dc  means  a  high  latent  heat. 
The  best  fluids  for  refrigeration  are  therefore  those  in  which  the  ratio  of 
latent  heat  to  specific  heat  of  liquid  is  large.  From  this  standpoint,  ammonia 
is  among  the  most  efficient  of  the  vapors  used.  With  carbon  dioxide, 
the  area  under  da  forms  a  large  deduction  from  the  gross  refrigerating 
effect. 

631.  Fluids  Used.  The  vapors  used  for  refrigeration  include  sulphuric  ether, 
sulphur  dioxide,  methylic  ether,  ammonia,  carbon  dioxide,  ethyl  chloride,  Pictet 
fluid  (a  mixture  of  carbon  dioxide  and  sulphur  dioxide),  and  steam.  The  vapor 
chosen  must  not  be  too  expensive,  and  it  must  not  exert  a  detrimental  influence  on 
the  machinery.  Ether,  once  commonly  employed,  is  quite  costly ;  its  specific  volume 
is  so  great  that  the  machines  were  excessively  bulky.  The  inward  leakage  of  air 
resulting  from  the  extremely  low  pressures  necessary  often  heated  the  compressor 
cylinder.  Sulphur  dioxide  unites  with  water  to  form  sulphurous  acid,  which  rapidly 
corrodes  the  cylinder  when  any  moisture  enters  the  system.  The  Pictet  fluid  has  been 
used  only  by  its  inventor.  Carbon  dioxide,  though  inefficient,  has  been  commer- 
cially satisfactory  excepting  where  its  low  critical  temperature  (Art.  379)  was 
objectionable.  Ammonia  is  the  fluid  principally  employed ;  the  only  serious  ob- 
jection to  it  seems  to  be  the  presence  of  occasional  traces  of  moisture.  The  ordi- 
nary ammonia  of  commerce  is  a  weak  aqueous  solution  of  the  gas,  H3N\  The 
ammonia  employed  in  refrigerating  machines  is  the  nearly  pure  anhydrous  lique- 
fied gas,  which  has  an  intensely  irritating  and  dangerous  odor.  It  boils  at  -26°  F. 
at  atmospheric  pressure. 


632.    Comparisons.    It  is  interesting  to  compare  the  effects  following  the  use 
d  of  various  fluids  between  assigned  temperature  limits. 

Let  the  cycle  be  one  in  which  the  vapor  is  dry  at 
the  beginning  of  compression,  abcde,  Fig.  309  b.  We 
have 

Q  =  ciefg  -  gabh  -  Le-  (lib  -  7>a). 


g    h 


t 


FIG.  3096.      Art.  632. —Dry 
Compression  Cycle. 

The  value  of  Td  is  Te(£* 
of  k  is  variable  ;  but  we  have 


W=q-Lt. 
q  =  gabcdf  =  gabh  +  libci  +  iccJf 


,  where  y  is  the  adiabatic  exponent.     The  value 


J?  =  ne-  nc,  or  k  log  Td-k  log  Tc  =  (ne  -  nc}  *  2.3, 


COMPARISON   OF  VAPORS 


407 


in  which  Td,  !TCT  nn  and  nc  are  known.     The  following  are  specimen  results  (see 
tables,  pp.  247,  248,  422,  424)  : 

NH3 

Tb  64.4°  F. 

Ta  5°  F. 

Lc  520.22 

Le  582.1 


ft. 

P  = 

Pe= 

Td 

k 

ne 

nc 

y 

<1 

Q 
w 


-  25.63 
117.42 
33.667 
175°  F. 
0.70 
1.20 
1.065 
1.33 
659.91 
519.61 
77.81 
6.68 


SO2 
64.4°  F. 

5°F. 
153.81 
169.745 
10.44 
-8.449 
44.537 
11.756 
159°  F. 
0.2023 
0.3478 
0.3140 
1.272 
191.8 
150.86 
22.05 
6.82 


H20 
116°  F. 
32°  F, 
1038 
1060 
84 
0 

1.5 

0.0886 
484°  F. 
0.493 
2.1832 
1.9412 
1.298 
1303 
976 
243 
4.02 


633.  Capacity.  The  common  basis  for  rating  refrigerating  machines 
is  in  tons  of  ice-melting  effect  per  24  hr.  The  "  ice-melting"  effect  is  a  con- 
ventional term  denoting  the  performance  of  142  B.  t.  u.  of  refrigeration. 
(The  latent  heat  of  fusion  of  ice  is  approximately  142  B.  t.  u.)  Let  Q  be 
the  heat  removed  from  the  vaporizer  per  cubic  foot  of  fluid  measured  at 
its  maximum  volume  during  the  cycle  ;  then  the  tonnage  per  cubic  foot  is, 

theoretically, 

T=Q  -(142x2000). 

Let  D  be  the  piston  displacement,  per  24  hr.,  in  cubic  feet ;  then  the  "  rat- 
ing "  of  the  machine  is 

t=DT=D+  284000. 


In  practice,  this  does  not  exactly  hold,  because  the  vapor  is  superheated 
by  the  cylinder  walls  during  the  suction  stroke,  its  density  being  thus 
decreased  below  that  of  the  saturated  vapor.  The  reduction  of  capacity 
due  to  this  superheating  may  be  represented  by  the  empirical  expression 
0.04  p  —  P,  in  which  p  is  the  pressure  in  the  condenser,  and  P  that  at  the 
vaporizer.  The  actual  tonnage  is  then 

-  0.04  £-}DQ  +  284000. 


634.    Economy.     A  practical  unit  of  economy  is  the  pounds  of  ice-melt^ 
ing  effect  per  pound  of  coal  burned  in  the  boiler  which  drives  the  com- 


408  APPLIED  THERMODYNAMICS 

pressor  engine.     The  refrigerating  effect  per  cubic  foot  of  fluid  is,  if  we 
ignore  self -evaporation  (Art.  625), 


the  work  done  in  the  compressor  cylinder  is  (q  —  Q) ;  that  in  the  engine 
cylinder  is  C(q—  Q),  in  which  C  is  the  reciprocal  of  the  combined  mechani- 
cal efficiency  of  engine  and  compressor,  ranging  from  1.15  to  1.25  for  direct 
connected  units.  The  foot-pounds  of  refrigerating  effect  per  foot-pound 
of  indicated  work  in  the  engine  cylinder  are  then 


The  ice-melting  effect  per  horse  power  hour  is  then 
1980000  A 


142  x  778V  P 

If,  as  in  ordinary  average  practice,  three  pounds  of   coal  are  used  per 
Ihp.-hr.,  the  ice-melting  effect  per  pound  of  coal  is 

1980000      (l_QMP_\Q+C( 
142  x3x  778V  P 

635.   Cooling  Water.    The  heat  absorbed  by  the  condenser  per  cubic  foot 
of  piston  displacement  is 

-0.04 


The  number  of  pounds  of  water  required  per  24  hr.  to  absorb  this  heat, 
assuming  the  temperature  rise  of  the  water  to  be  30°,  is 

A  -0.04  £\Dq-t-30. 

The  gallons  of  water  necessary  per  minute  for  each  ton  of  "  rating  "  (as 
defined  in  Art.  633)  then  become, 


1.0- 0.04       Dg-h30  x  60  x  24  x  S$ -h     l  -  Q 


/142x2000>\|. 


This  is  about  one  gallon  for  the  given  range  of  water  temperature ;  the 
usual  range,  however,  is  only  about  15°. 

636.    Size  of  Compressor.     If  the  fluid  at  the  beginning  of  compression 
be  just  dry,  and  v  be  the  specific  volume  and  M  the  weight  of  this  dry 


COMPRESSOR  DESIGN  409 

vapor  circulated  per  minute,  the  total  volume  displaced  per  minute  is  Mv; 
if  N  be  the  number  of  single  strokes  per  minute,  the  piston  displacement 
per  single  stroke  of  a  double-acting  compressor  must  be  D  =  Mv  -5-  N. 
This  must  be  increased  for  superheating,  as  in  Art.  633,  the  displace- 
ment becoming 


-- 

and  must  be  further  corrected  for  clearance,  as  in  Art.  233.  A  small 
additional  increase  is  made  in  practice,  to  allow  for  the  presence  of  air 
and  moisture,  etc. 

637.  Compressor  Design.     The  refrigerating  effect  being  assigned,  the  nor- 
mal (un  refrigerated)  vaporizer  temperature  and  the  possible  condenser  tempera- 
ture are   ascertained.     These  determine  the  cyclic  limits.     The  type   (single-  or 
double-acting)  and  rotative  speed  of  the  compressor  are  then  fixed.     The  refriger- 
ating effect  per  pound  of  fluid  under  the  assumed  temperature  conditions  is  now 
computed,  and  the  necessary  weight  of  fluid  determined.     The  piston  displace- 
ment may  then  be  calculated  and  the  power  consumption  and  cooling  water  supply 
ascertained. 

In  most  vapor  computations,  the  specific  volume  of  the  liquid  may  be  ignored. 
This  does  not  hold  with  carbon  dioxide,  which  is  worked  so  near  its  critical  tem- 
perature that  the  specific  volume  of  the  liquid  closely  approaches  that  of  the  vapor. 
The  losses  in  the  vapor  compressor  are  similar  in  nature,  though  opposite  in  effect, 
to  those  in  the  steam  engine  cylinder.  The  transfer  of  heat  between  cylinder  walls 
and  working  fluid  causes  the  most  serious  loss  ;  it  is  to  be  overcome  in  the  same 
ways  as  are  employed  in  steam  engine  practice. 

638.  Steam  Compressors.     In  these,  the  working  fluid  is  water,  injected  at 
ordinary  temperature  into  a  vacuum  chamber.     A  portion  of  the  water  vaporizes, 
absorbing  heat  from  the  remainder  and  thus  chilling  it.    The  vapor  is  then  slightly 
compressed,  condensed,  and  pumped  away  or  back  to  the  vaporizer.     The  principle 
of  action  is  the  same  as  that  of  any  vapor  machine,  but  the  pressure  throughout 
is  less  than  that  of  the  atmosphere.     The  temperature  cannot  be  lowered  below 
32°  F.  (Art.  632). 

639.  Ammonia  Absorption  Machine.    This  was  invented  by  Carre.    The 
theory  has  been  thoroughly  presented  by  Ledoux  (5)  ;  numerous  develop- 
ments of  the  original  Carre  apparatus  have  been  described  by  Wallis- 
Tayler  (6)  .    Instead  of  using  the  mechanical  force  exerted  by  a  compressor 
to  raise  the  temperature  of  the  fluid  emerging  from  the  vaporizer,  this 
elevation  of  temperature  is  produced  by  the  application  of  external  heat 
from  fuel  or  steam  coils  in  what  is  called  the  generator.     The  fluid  then 
passes  to  the  condenser,  and  through  an  expansion  valve  to  the  vaporizer. 
It  cannot  be  returned  directly  to  the  generator,  because  the  pressure  there 
exceeds   that   in   the   vaporizer.     An   intermediate    element,  called    the 


410 


APPLIED   THERMODYNAMICS 


absorber,  is  used.  The  operation  depends  upon  the  well-known  fact  that 
water  has  the  power  of  dissolving  large  volumes  of  volatile  vapors ;  at 
59°  F.,  it  dissolves  727  times  its  own  volume  of  ammonia.  This  solution 
produces  an  exothermic  reaction;  heat  is  evolved,  amounting  to  about 
926  B.  t.  u.  per  pound  of  vapor  absorbed.  "  The  mechanical  force  which 
draws  the  vapor  from  the  vaporizer  in  the  compression  system  is  here  re- 
placed by  the  affinity  of  water  for  ammonia  vapor ;  and  the  mechanical 
force  required  for  compressing  the  vapor  is  replaced  by  the  heat  of  the 
generator,  which  severs  this  affinity  and  sets  the  vapor  at  liberty  "  (Kent). 
Ammonia  is  among  the  most  soluble  of  the  substances  considered ;  other 
vapors  may,  however,  be  used  (7). 

640.  Arrangement  of  Apparatus.  The  absorption  apparatus  is  shown 
in  outline  in  Fig.  310.  At  A  is  the  generator,  containing  a  strong  solution 
of  ammonia  in  water  and  suitably  heated.  The  heat  liberates  ammonia 
gas,  which  passes  through  the  pipe  a  to  the  conlenser  B.  From  this  the 
liquefied  ammonia  passes  out  at  6  and  is  expanded  through  the  valve  7i, 
taking  up  heat  from  the  vaporizer  (7,  as  in  the  compression  system.  The 


FIG.  310.    Art.  640.  —  Ammonia  Absorption  Apparatus. 

absorber  D  is  a  vessel  containing  water  or  a  weak  solution  of  ammonia  in 
water.  The  solution  of  vapor  in  this  water  produces  a  suction  which  con- 
tinually draws  vapor  over  from  C  to  D.  The  solubility  of  ammonia  in 
water  decreases  as  the  temperature  increases,  so  that  the  evolution  of  heat 
in  the  absorber  must  be  counteracted  by  jacketing  that  vessel  with  water 
or  installing  water  coils  in  the  solution.  The  waste  water  from  the  con- 
denser may  be  used  for  this  cooling.  The  more  concentrated  portion  of 
the  liquid  in  D  is  now  pumped  through  /  to  A,  while  the  weaker  solution 
is  drawn  off  from  the  bottom  of  A  and  returned  to  the  top  of  D  through  d. 
A  coil  heater  at  E  provides  for  the  interchange  of  heat,  thus  warming  the 
liquid  entering  A  and  cooling  that  entering  D,  as  is  to  be  desired. 

641.    Cycle.     From  the  condenser  to  the  vaporizer,  the  operation  is 
identical  with  that  in  a  compression  plant.     The  absorber  and  generator 


ABSORPTION  APPARATUS  411 

replace  the  compressor.  The  rise  in  pressure  occurs  between  the  punip  / 
and  the  generator  outlet  a.  In  Fig.  311,  B  may  be  taken  as  the  state  of 
the  gas  entering  the  condenser,  in  which  it  is  liquefied  along  BA.  Ex- 
pansion reduces  its  pressure,  giving 
the  path  AJ.  In  passing  through  the 
vaporizer,  the  liquid  is  evaporated 
along  JC.  It  cannot  be  returned  di- 
rectly to  the  generator;  nor  can  it 
advantageously  be  returned  by  pump- 
ing, because  very  little  solution  would 
occur  at  the  high  temperature  main- 
tained in  the  generator.  It  is  there- 
fore absorbed  by  water  in  D,  Fig.  310, 


at  a  pressure  nearly  equal  to  that  in      FIG.  31 1.    Arts,  oti,  642.—  Absorption 
C,  and  transferred  to  the    generator,  Cycle, 

where  its  pressure  rises,  as  along  CB,  Fig.  311.  From  C  to  B,  the  vapor 
is  in  solution;  but  its  pressure  and  temperature  are  increased  by  the 
application  of  heat,  just  as  in  compression  machines  they  are  increased  at 
the  expenditure  of  external  work.  The  cycle  is  the  same  as  that  of  the 
compressive  apparatus. 

642.    Comparison  of  Systems.     The  temperature  attained  at  B,  Fig.  311,  is 

practically  the  same  as  in  dry  compressive  system  s ;  it  is  TB  =  Tc  (  ~ )  V  =  Tc(  ^}  °'2 

V/W  \PC' 

for  ammonia  (y  =  1.33).  The  refrigeration  per  pound  of  pure  dry  vapor  is 
Q  =  (1  —  A')Z,  as  with  the  compressor.  Ideally,  the  heat  evolved  in  the  absorber 
should  be  approximately  sufficient  to  evaporate  the  solution  in  the  generator. 
Actually,  this  heat  is  largely  lost,  on  account  of  the  necessity  of  cooling  the  ab- 
sorber. Assuming  that  all  the  steam  consumed  by  the  pumps  is  afterward  em- 
ployed in  the  generator,  the  heat  consumption  of  the  absorption  apparatus  includes 
the  following  four  items: 

72,  that  necessary  to  evaporate  the  cold  water  drained  back  from  a  portion  of 
the  condenser  tubes ; 

E,  that  necessary  to  raise  the  temperature  of  the  solution  entering  the  genera- 
tor to  that  of  saturation  ; 

S,  that  necessary  to  distill  the  ammonia  in  the  generator  (latent  heat  plus  heat 
of  decomposition) ; 

W,  necessary  to  raise  the  temperature  of  the  vapor  during  superheating. 

Symbolically,  H  =  W  +  S  +  E  +  R.  Items  E  and  R  may  be  regarded  as  off- 
set by  the  friction  losses  in  the  compressor  system.  We  may  then  put  H=  W+  S 
in  the  absorption  system.  "  A  rough  comparison  of  the  two  systems  is  as  follows : 
At  a  suction  pressure  of  about  34  ll>.  absolute,  at  which  the  vaporizer  temperature 
is  5°  (with  ammonia),  a  good  non-condensing  steam  engine  will  consume  heat 
amounting  to  about  969  B.  t.  u.  per  pound  of  ammonia  circulated,  the  condenser 
temperature  being  65°.  Under  the  same  conditions,  the  absorption  machine  will 


412  APPLIED  THERMODYNAMICS 

consume  about  72  B.  t.  u.  in  raising  the  temperature  and  about  897  B.  t.  u.  in  dis- 
tilling the  ammonia ;  whence  H  =  72  +  897  =  969.  The  two  machines  are  thus 
equal  in  economy  for  a  suction  pressure  of  34  Ib."  As  the  vaporizer  temperature 
falls  below  5°,  the  economy  of  the  absorption  system  becomes  better  than  that  of 
a  compressor  with  a  non-condensing  engine.  The  reverse  is  the  case  when  the 
vaporizer  temperature  rises.  Compared  with  condensing  engine  driven  compres- 
sors, the  economy  is  about  equal  for  the  two  types  when  the  vaporizer  room  tem- 
perature is  zero.  Where  a  low  back-pressure  is  required,  as  in  ice-making,  the 
absorption  system  is  thermodynamically  superior. 

643.  Steam  Absorption  Machines.     A  water-vapor  machine  of  the  class  de- 
scribed in  Art.  638  may  dispense  with  the  compressor,  the  steam  being  absorbed 
by  and  generated  from  solutions  in  sulphuric  acid.     This  form  of  apparatus  has 
been  in  use  for  at  least  a  century,  having  been  successfully  developed  by  Carre  and 
others  (8). 

DETAILS  AND  COMMERCIAL  STANDARDS 

644.  Direct  Expansion.     When  the  refrigerating  fluid  is  itself  circu- 
lated in  the  room  or  through  the  material  to  be  cooled,  the  system  is  that 
of  direct  expansion.     While  simple  and  economical,  there  are  objections  to 
this  type  of  plant.     The  least  movement  of  the  expansion  valve  changes 
the  lower  pressure  and  temperature,  and  consequently  the  temperature  of 
the  room  to  be  cooled.     The  introduction  of  a  substance  like  ammonia  is 
often  considered  too  hazardous  in  rooms  where  valuable  materials  like 
furs  would  be  damaged  by  any  leakage. 

645.  Brine  Circulation.     By  expanding  the   refrigerating  fluid  in  coils  im- 
mersed in  some  harmless  liquid,  like  salt  water,  the  former  may  be  kept  wholly 
within  the  power  plant ;  the  cooled  water  is  then  circulated  through  the  rooms  to 
be  refrigerated  by  means  of  a  pump.     The  operation  is  wasteful,  because  it  in- 
volves an  irreversible  rise  in  temperature  between  working  fluid  and  brine,  but 
is  often  preferred  for  the  reasons  given.     The  brine  serves  as  a  "fly  wheel  for 
heat,"  smoothing  out  the  variations  in  temperature  which  occur  with  direct  expan- 
sion ;  but  a  secondary  circulating  system  is  more  expensive  in  installation   and 
operation.    In  addition  to  the  usual  apparatus,  there  must  be  supplied  a  brine  tank, 
which  now  becomes  the  vaporizer,  coils  within  the  brine  tank,  and  a  brine  pump. 
The  cooling  coils  in  the  refrigerated  room,  and  the  piping  thereto,  must  be  sup- 
plied as  in  direct  expansion  ;  they  are,  however,  rather  less  expensive. 

646.  Fluids.     Salt   brine   is   commonly  used    rather   than    water,  since    the 
freezing  point  of  the  former  may  be  as  low  as  -  5°  F.     This  fluid  is  detrimental 
to  cast-iron  fittings,  and  these  are  ordinarily  made  extra  heavy  when  used  for 
brine  circulation.     Chloride  of  calcium  in  solution  permits  of  a  still  lower  tempera- 
ture ;  it  may  solidify  at  as  low  a  temperature  as  —  54°  F.     A  solution  of  magne- 
sium chloride  is  occasionally  used.     Salt  brine  cannot  be  left  in  the  system  after 
the  circulation  ceases,  as  the  salt  settles  out  and  the  freezing  point  is  raised. 


APPLICATIONS  OF   REFRIGERATION  413 

647.  Brine  Circulation  Plant.     Figure  312  shows  a  complete  plant.     In  opera- 
tion, the  compressor  is  first  started,  drawing  the  air  out  of  the  pipe  coils.     A  drum 
of  anhydrous  ammonia  is  placed  at  B,  and  the  contents  allowed  to  run  into  the 
liquid  receiver  through  the  valve  C.     The  expansion  valve  D  is  then  opened  and 
liquid  ammonia  passes  through  to  the  brine  tank.     The  valves  A  and  F  are  kept 
open  until  the  odor  of  ammonia  is  evident.     They  are  then  closed,  the  valve  L  is 
opened,  and  the  water  turned  on  at  the  condenser.     The  compressed  vapor  is  now 
liquefied  in  the  condenser,  its  temperature  falling  within  20°  of  that  of  the  cool- 
ing water  in  usual  practice.     The  brine  pump  G  is  started,  circulating  the  chilled 
brine  through  the  refrigerated  room  H,  and  the  speed  of  the  compressor  is  in- 
creased until  the  temperature  of  the  fluid  in  the  brine  tank  is  about  20°  below 
the  required  temperature  in  H.     Ammonia  is  supplied  at  C  until  the  level  in  the 
receiver  remains  constant.     The  supply  is  then  cut  off.     At  the  beginning  of  the 
operation,  all  of  the  ammonia  will  be  evaporated  in  E,  and  the  vapor  will  be  highly 
superheated  during  compression.     As  the  brine  is  chilled,  the  temperature  of  the 
discharged  vapor  falls,  and  frost  forms  on  the  outside  of  the  pipe  /,  gradually 
approaching  the  compressor.     If  the  supply  of  fresh  liquid  is  stopped  at  this  point, 
superheating  will  continue  to  occur,  producing  "dry"  compression.     In  "wet" 
compression,  the  compressor  inlet  becomes  heavily  frosted  and  the  outlet  pipe  is 
sufficiently  cool  to  be  touched  by  the  hand.     With  adequate  jacketing,  etc.,  the 
dry  system  may  be  in  practice  as  economical  as  the  wet  (Art.  629),  but  additional 
care  is  necessary  to  avoid  leakage  at  the  stuffing  boxes.     A  direct  expansion  system 
has  already  been  shown  in  Fig.  306. 

648.  Indirect  Refrigeration.     In  some  cases,  neither  brine  circulating  coils 
nor  direct  expansion  coils  are  used  in  the  cooling  room,  but  air  is  blown  over  a 
bank  of  coils  and  thence  through  ducts  to  the  room.     This  constitutes  indirect 
refrigeration,  providing  ventilation   as  well  as  cooling.     In  direct  refrigeration, 
provision  is  sometimes  made  for  drawing  off  foul  air  by  vertical  flues.     In  certain 
applications,  arrangements  are  made  for  washing  or  filtering  and  drying  the  air 
supply  introduced. 

649.  Abattoirs.  Packing  Houses.     Refrigeration  here  plays  an  important  part. 
Either  direct  expansion  or  brine  circulation  may  be  employed,  the  coils  being 
located  along  the  side  walls  near  the  ceiling,  or  suspended  from  the  ceiling,  if 
head  room  will  permit.     The  latter  is  the  better  arrangement.     Moisture  from 
the  atmosphere  of  the  room  rapidly  condenses  on  the  outside  of  the  pipes,  and 
provision  must  be  made  for  removal  of  the  drip.     The  atmosphere  of  the  room 
rapidly  becomes  dry. 

650.  Cold  Storage.     For  preserving  vegetables,  fruits,  poultry,  eggs,  butter, 
milk,  cheese,  fish,  meats,  etc.,  either  in  permanent  storage  or  during  transporta- 
tion, mechanical  refrigeration  has  been  widely  applied.     Temperatures  of  from 
25°  to  40°  F.  are  usually  maintained,  the  temperature  being  lowered  gradually. 
Some  substances  keep  best  when  actually  frozen.    Mechanically  cooled  refrigera- 
tor cars  have  been  described  by  Miller  (9).     For  all  storage-room  applications, 
the  fundamental  principles  underlying  the  computation  of  the  amount  and  dis- 


411 


APPLIED  THERMODYNAMICS 


\\\\\\\\\\\\\\\\\\\\\\\^^ 


V\\\\\\\\\\\\\\\\\\\\\\\W 


ICE  MAKING  415 

tribution  of  coil  surface  are  precisely  those  employed  in  the  design  of  heating  and 
ventilating  systems.  Reference  should  be  made  to  the  works  of  Siebel  (10)  and 
Wallis-Tayler  (11).  The  thorough  insulation  of  the  rooms  and  of  the  conduct- 
ing pipes  is  of  much  importance. 

651.  Other  Applications.     Mechanical  refrigeration  is  universally  employed  in 
breweries,  for  cooling  of  the  cellars  and  the  wort,  as  well  as  for  cooling  during 
fermentation  (attemperator  system)  (12).     It  is  popular  in  marine  service,  where 
the  space  occupied  by  stored  ice,  and  its  shrinkage,  would  be  serious  items  of 
expense.     It  is  applied  in  candy  factories,  for  cooling  chocolate;  in  candle  and 
paraffin  works  and  linseed-oil  refineries  for  precipitating  out  solid*  waxes  from 
mixtures;  in  dairies  for  cooling  the  milk;  in  tea  warehouses,  dynamite  factories, 
in  the  manufacture  of  photographic  dry  plates,  in  wine  cellars,  soda-water  estab- 
lishments, sugar  refineries,  chemical  works,  glue  factories,  and  for  the  winter  stor- 
age of  furs.    The  losses  experienced  in  marine  transportation  of  cattle  on  the  hoof 
have  been  greatly  reduced  by  cooling  the  space  between  decks.     Refrigeration  has 
also  been  used  for  congealing  quicksand  during  excavation  and  tunneling  opera- 
tions in  loose  soil. 

A  recent  application  is  in  the  formation  of  indoor  skating  ponds.  These  are 
frozen  by  direct  expansion  through  submerged  coils.  A  fresh  surface  is  frozen 
on  whenever  necessary,  and  this  is  kept  smooth  by  the  use  of  a  planing  machine. 
Pipe-line  refrigeration  from  central  stations  is  being  practiced  in  at  least  nine 
American  cities;  the  present  status  of  this  public  service  has  been  studied  by 
Hart  (13). 

652.  Ice  Making.     This  is  one  of  the  most  important  applications.     The 
manufacture  of  ice  may  be  carried  on  as  an  adjunct  to  the  ordinary  operation  of 
any  refrigerating  plant.     The  product  is  from  an  hygienic  standpoint  immeasur- 
ably superior  to  the  usual  natural  ice.     In  practice,  three  systems  are  used :  the 
plate,  the  stationary  cell,  and  the  can,  the  last  being  of  most  importance. 

653.  Plate  System.   Large,  shallow,  hollow,  rectangular  boxes  are  immersed  in 
a  tank  containing  the  water  to  be  frozen,  dividing  the  body  of  water  into  narrow 
sections,  corresponding  to  the  "  plates  "  of  ice  to  be  formed.     Through  the  hollow 
boxes,  a  solution  of  chilled  brine  circulates ;  in  some  cases,  however,  this  brine  is 
quiescent,  being  chilled  by  coils  immersed  in  it,  in  which  coils  brine  from  the 
compressor  plant  circulates.     A  "  plate  "  14  in.  thick  may  be  produced  in  from 
9  to  14  days.     The  plates  when  formed  are  loosened  by  circulating  warm  brine  for 
a  few  moments,  and  are  then  hoisted  out  by  cranes. 

654.  Stationary  Cell  System.     A   large  number  of   approximately  cubical 
tanks,  with  hollow  walls  and  bottoms,  are  set  in  a  frame.     Brine  is  circulated 
through  the  walls.     A  "  cake  "  of  ice  is  gradually  formed  within  the  tanks.     This 
is  loosened  in  the  same  manner  as  plate  ice. 

655.  Clear  Ice.     Much  difficulty  has  been  experienced  in  securing  a  product 
free  from  the  characteristic  porous,  granular   structure.     A  clear   ice  has  been 


416  APPLIED  THERMODYNAMICS 

found  to  be  most  probable  when  the  temperature  of  the  operation  is  not  too  low, 
when  the  water  is  agitated  during  cooling,  and  when  the  layers  are  thin,  as  in  the 
plate  system  or  with  shallow,  stationary  cells.  To  provide  these  conditions  usually 
involves  delay,  trouble,  or  expense.  The  clear  ice  of  the  present  day  is  pro- 
duced by  the  use  of  distilled  water.  This  may  be  obtained  by  condensing  the 
exhaust  from  the  compressor  engine,  or  by  using  that  exhaust  in  an  evaporator  to 
distill  in  vacuo  a  fresh  supply  of  water.  Traces  of  cylinder  oil  must  in  the  former 
case  be  thoroughly  eliminated,  and  the  water  carefully  filtered. 

656.  Can   System.     The  use  of  distilled  water  from  the  engine  exhaust  in 
portable  cans  is  at  present  standard  practice.     The  cans,  of  plain  galvanized  iron, 
stand  in  a  tank  containing  a  circulating  solution  of  brine,  the  temperature  being 
somewhat  below  32°.     Blocks  of  300  Ib.  weight  are  produced  in  from  50  to  60  hr. 
—  about  one  fourth  the  time  usually  required  with  the  plate  system.     The  ice  is 
loosened  by  lowering  the  cans  for  a  moment  in  warm  water.     The  various  wastes 
of  water,  when  the  condensation  from  the  engine  is  employed,  require  that  the 
amount  fed  the  boiler  shall  be  about  33  per  cent  in  excess  of  the  amount  of  ice  to 
be  made.     A  highly  economical  steam  engine  is  thus  undesirable.     "  The  can  sys- 
tem requires  about  one  fourth  the  floor  area  and  one  twelfth  the  cubical  space  that 
are  needed  by  the  plate  system  for  the  same  output,  while  it  is  about  four  times 
as  rapid,  and  costs  initially  about  25  per  cent  less." 

In  a  system  recently  introduced,  large  hollow  cylinders,  through  which  ammonia 
circulates,  are  revolved  in  a  freezing  tank.  A  thin  film  of  ice  forms  on  the  outside 
of  the  cylinders,  and  is  scraped  off  by  knives  and  pumped  in  slushy  condition  to  a 
hydraulic  press,  where  it  is  formed  into  cakes.  The  process  is  continuous  and  re- 
quires little  labor.  The  clearness  of  the  ice  depends  upon  the  pressure  to  which 
it  is  subjected. 

657.  Details.     The  pressure  range  is  usually  from  190  to  15  Ib.  gauge  approxi- 
mately.    The  brine  may  be  ordinary  salt  brine,  consisting  of  3  Ib.  of  medium 
ground  salt  per  gallon  of  water  (specific  heat  about  0.8),  or  calcium  chloride  brine, 
in  the  proportion  of  3  to  5  Ib.  of  chloride  to  one  gallon,  or,  on  the  average,  at  about 
23°  Be.,  weighing  13|  Ib.  per  gallon  and  permitting  of  a  temperature  of   —9°  F. 
The  specific  heat  of  this  solution  is  about  0.9.     The  brine  must  be  periodically 
examined  with  a  salinometer.     The  ice-making  capacity  is  not  the  same  as  the  ice- 
melting  effect  described  in  Art.  633.     To  produce  actual  ice,  the  water  must  be 
cooled  from  its  initial  temperature  to  the  freezing  point,  while  the  ice  is  usually 
formed  at  a  temperature  considerably  below  32°.     Roughly  speaking,  about  one- 
half  ton  of  actual  ice  may  be  made  per  ton  of  rated  capacity.     The  productive 
capacity  is  further  reduced  by  the  losses  attending  the  handling  of  the  ice. 

658.  Tonnage  Rating.     The  ice-melting  effect  of  a  machine  work- 
ing between  the  pressures^  and  P  is,  from  Art.  633, 


t  =  ml  1  -  0.04  jUPGI  -  JT)£  ^-  (142  x  2000), 
in  which  m  is  the  density  of  the  vapor  at  the  suction  pressure. 


TONNAGE  RATING  417 

Since  X  is  determined  by  p  and  P,  the  capacity  depends  directly 
upon  the  pressure  range  and  the  piston  displacement.  The  Ameri- 
can Society  of  Mechanical  Engineers  (14)  has  standardized  these 
pressures  by  assigning  90°  and  0°  F.  as  the  corresponding  tempera- 
ture limits.  This  makes  the  lowest  possible  room  temperature  about 
15°  F.  with  direct  expansion  and  about  25°  F.  with  brine  circulation. 
Lower  temperatures  are  frequently  required.  The  lower  of  the 
assigned  temperatures  also  fixes  the  value  of  m.  For  any  other 
pressures,  q,  Q,  at  the  state  M,  x,  ?,  the  tonnage  capacity  would  be 

T=  M(\  -  0. 04  I- W1  -  x)l+  (142  x  2000) ;  whence 
\  Q/ 


T 

t        ^M      o.04 


P 

659.  Compressor  Proportions.     The  builders  of  machinery  do  not  in  all 
cases  rate  their  machines  on  this  basis.     Many  of  them  merely  state  the 
piston  displacement  (which  may  range  from  6500  to  8700  cubic  inches  per 
minute  per  ton  of  nominal  capacity)  or  the  weight  of  vapor  circulated 
under  given  pressure  conditions.     Power  rates  usually  range  from  one  to 
two  horse  power  at  the  engine  per  ton  of  capacity;  piston  speeds  vary 
from  125  to  600  ft.  per  minute. 

660.  Tests.     A  standard  code  for  trials  of  refrigerating  machines 
is  under  consideration  by  a  committee  of  the  American  Society  of 
Mechanical   Engineers,   a  preliminary   report   having   already  been 
made  (15).     Results  of  tests  are  stated  in  ice  melting  effect  in  pounds 
per  pound  of  coal  or  per  indicated  horse  power  hour  at  the  compressor 
engine.     Where  the  coal  is  not  measured,  3  Ib.  of  coal  per  hour  are 
often  assumed  to  be  equivalent  to  one  horse  power.     Let  a  be  the 
ice  melting  effect  per  indicated  horse  power:  then 

142  a  -  (1980000  -  778)  =  0.0557  a 

is  the  efficiency  from  engine  cylinder  to  cooling  room.      Let  b  be  the 
ice-melting  effect  per  pound  of  coal  containing  14,000  B.  t.  u. ;  then 

142  b  -14000  =  0.01015  b 

is  the  efficiency  from  coal  to  cooling  room.     A  few  well-known  tests 
will  be  quoted. 


418  APPLIED  THERMODYNAMICS 

661.  Air  Machines.   Ledoux  quotes  tests  (16)  in  which  the  ice-melting  effect 
per  pound  of  coal  was  from  3.0  to  3.42  at  3  Ib.  coal  per  Ihp. ;  the  efficiencies  from 
coal  to  cooling  room  being  respectively  only  0.0304  and  0.0346.     A  Bell-Coleman 
machine  at  Hamburg,  tested  by  Schroter  (17)  gave  from  354  to  371  calories  of 
refrigeration  per  Ihp.-hr.,  the  efficiency  from  the  engine  to  cooling  room  being 
therefore  from  0.551  to  0.580.     The  range  of  temperatures  was  very  low.     About 
half  the  power  expended  in  the  compressor  is  ordinarily  recovered  in  the  expan- 
sion cylinder. 

662.  Compression  Machines.    Most  tests  have  been  made  with  ammonia. 
Ledoux  tabulates  (18)  ice-making  effects  per  pound  of  coal  ranging  from 
9.86  to  46.29,  based  on  3  Ib.  of  coal  per  horse  power ;  the  corresponding 
efficiencies  being  from  0.10  to  0.469.     A  number  of  tests  by  Schroter  gave 
from  19.1  to  37.4  Ib.,  or  from  0.194  to  0.379  efficiency.     Shreve  and  Anderson 
obtained  21  Ib.,  or  0.213  efficiency  (19).     Anderson  and  Page  (20)  obtained 
18.261  Ib.  of  ice-melting  effect  per  pound  of  coal  containing  12,229.6  B.  t.  u. 
per  pound ;  or  65.79  Ib.  per  Ihp.     The  efficiency  from  engine  to  refrigera- 
tion was  3.65;  from  coal,  it  was  0.211.     The  pressure  range  was  from 
28.88  to  132.01  Ib.  absolute.     Denton  (21)  reported  23.37  Ib.  of  ice-melt- 
ing effect  per  pound  of  coal  on  the  3  Ib.  basis,  working  between  27.5  and 
161  Ib.  pressure.     The  ice-melting  capacity  for  24  hr.  was  74.8  tons,  the 
average  steam  cylinder  horse  power,  85 ;  whence  the  engine  to  room  effi- 
ciency was  (23.37  x  3  x  142)  -^-2545  =  3.92,  and  the  coal  to  room  efficiency 
about   0.236.      The    efficiency   from   coal   to   engine   cylinder   was   then 
0.236-^-3.92  ==  0.0602.     A  series  of  tests  by  Schroter  (22) 'gave  from  1674 
to  4444  calories  of  refrigeration  per  compressor  horse  power,  the  corre- 
sponding  efficiencies   being  therefore  from  2.61  to  6.91 ;    the   engine  to 
room  efficiency  might  be  15  per  cent  less,  say  from  2.21  to  5.87.     A  Pictet 
fluid   machine  (23)   gave   3507   calories   per   horse  power  in  the  steam 
cylinder,  or  5.5  efficiency.     The  reason  for  these  high  values,  exceeding 
unity,  "has  been  stated  in  Art.  621.     The  steam  engine  efficiencies  in  none 
of  these  tests  exceeded  15  per  cent ;  it  did  not  average  much  over  5  per 
cent ;  an  average  efficiency  of  0.237  from  coal  to  room  corresponds  to  a 
coefficient  of  performance  of  about 

0.237 -=-0.05=4.74  (neglecting  friction  of  mechanism). 

The  engine  to  room  efficiency  is  equal  to  the  actual  coefficient  of  perform- 
ance multiplied  by  the  mechanical  efficiency  of  engine  and  compressor. 

663.  Ammonia  Absorption  Machines.     Assuming  an  evaporation  of  11.1  Ib.  of 
water  from  and  at  212°  F.  per  pound  of  combustible,  Ledoux  (24)  reports  a  test 
in  which  20.1  Ib.  of  ice-melting  effect  were  produced  per  pound  of  coal,  the  over- 
all efficiency  being  thus  0.204.     A  seven-day  test  by  Denton  (25)  gave  17.1  Ib., 
based  on  10  Ib.  of  steam  per  pound  of  coal,  the  corresponding  efficiency  being 
about  0.173.     The  pressure  range  was  from  23.4  to  150.77  Ib.  absolute.     The  tern- 


AMMONIA  COMPRESSOR 


419 


420  APPLIED  THERMODYNAMICS 

perature  range  was  from  272°  to  80°  F. ;  the  coefficient  of  performance  for  the 
Carnot  cycle  would  have  been  2.83.  The  equivalent  efficiency  from  coal  to  com- 
pressor cylinder  in  a  compression  machine  must  then  have  been  at  least 

0.173  -  2.83  =  0.0613 ; 
or  from  coal  to  engine  cylinder,  about 

1.2  x  0.0612  =  0.07344. 

664.  Commercial  Types.  Compressors  may  be  driven  directly  from  a  steam 
cylinder,  or  by  belt.  Any  form  of  slow-speed  engine  maybe  used  for  driving; 
a  favorite  arrangement  is  to  have  the  steam  cylinder  horizontal  and  the  ammonia 
cylinder  vertical,  as  in  Fig.  313.  Tandem  or  cross-compound  engines  may  be 
used.  The  ammonia  condenser  may  be  an  ordinary  surface  condenser,  or  an 
atmospheric  condenser  of  the  form  described  in  Art.  585,  consisting  of  a  coil  of 
exposed  pipes  over  which  streams  of  water  trickle.  In  other  types,  the  ammonia 
coils  are  submerged  in  a  tank  of  circulating  water.  Cooling  towers  are  used 
where  there  is  an  inadequate  water  supply. 

(1)  Wallis-Tayler,  Refrigeration,  Cold  Storage,  and  Ice  Making,  1902.  (2)  Zeuner, 
Technical  Thermodynamics  (Klein  tr.),  I,  384.  (3)  Op.  cit.  (4)  Proc.  Inst.  Mech. 
Eng.,  1881,  105;  1886,  201.  (5)  Ice-making  Machines,  D.  Van  Nostrand  Co.,  1906. 
(6)  Op.  cit.,  p.  154  et  seq.  (7)  Wallis-Tayler,  Op.  cit.,  p.  25.  (8)  Wallis-Tayler,  op. 
cit.,  pp.  24-32.  (9)  Stevens  Indicator,  April,  1904  ;  Railroad  Gazette,  October  23, 
1903.  (10)  Compend  of  Mechanical  Refrigeration.  (11)  Op.  cit.  (12)  Op.  cit.,  381. 
(13)  Engineering  Magazine,  June,  1908,  p.  412.  (14)  Transactions,  1904.  (15)  Trans- 
actions, XXVIII,  8,  1249.  (16)  Ice-making  Machines,  1906,  Table  A.  (17)  Unter- 
suchungen  an  Kaltemachinen,  1887.  (18)  Loc.  cit.  (19)  Wood,  Thermodynamics, 
1905,  352.  (20)  Ibid.,  348.  (21)  Trans.  A.  8.  M.  E.,  XII.  (22)  Peabody,  Thermo- 
dynamics, 1907,  414.  (23)  Schroter,  Verg.  Vers.  an  Kaltemaschinen.  (24)  Loc.  cit. 
(25)  Trans.  A.  S.  M.  E.,  X. 

SYNOPSIS   OF   CHAPTER   XVIII 

A  heat  cycle  may  be  reversed,  the  heat  rejected  exceeding  that  absorbed  by  the  ex- 
ternal work  done. 

The  Carnot  cycle  would  lead  to  a  bulky  machine.  Actual  air  machines  work  with  a 
regenerator  or  in  the  Joule  cycle.  In  this  latter,  the  low-temperature  heat  ex- 
tracted from  the  body  to  be  cooled  is  mechanically  raised  in  temperature  so  that 
it  may  be  carried  away  at  a  comparatively  high  temperature.  The  mechanical 
compression  may  occur  in  one  or  more  stages. 

The  Joule  cycle  is  bounded  by  two  constant  pressure  lines  and  two  like  polytropics. 
If  the  latter  are  adiabatics, 

W=  Mk(Tb  -Tc-Ta  +  To),  q  =  Mk(Tc  -  Td~),  q  =  Mk(Tb  -  2Ta). 
The  displacement  per  stroke  of  a  double-acting  compressor  is  METC  -=-  2  NPC ;  that  of 
the  engine  is  MRTd  +  2NPd;  the  two  displacements  ordinarily  have   the  ratio 

T 

— £•    These  are  to  be  modified  for  clearance,  etc. 
Td 

Open  type  air  machines  work  between  pressures  of  14.7  and  70  to  85  Ib. ;  "dense  air 
machines'1'1  between  65  and  225  Ib.,  using  closed  circulation  and,  in  some  cases,  a 
regenerator. 


MECHANICAL  REFRIGERATION  421 

Coefficient  of  performance  =•§-'>  its  value   usually  exceeds  unity  ;   the  temperature 

T  T 

range  should  be  low.    Value  for  Joule  cycle  =  — =4 —  =  — =J — ,  if  paths  are  adi- 
abatic.  Ta-Td      T>  -  Tc 

The  Kelvin  warming  machine  works  in  the  Joule  cycle  and  delivers  heat  proportional 

T                 TV. 
to  the  work  expended  in  the  ratio —  = — ,  which  may  greatly  exceed 

unity.  2a  ~'ld      Tb~  7c 

The  vapor  compression  machine  uses  no  expansion  cylinder.     Refrigeration  results  from 

evaporation,  but  is  reduced  by  the  excess  liquid  heat  carried  to  the  cold  chamber. 
The  vaporizer  is  the  body  to  be  cooled  ;  the  condenser  removes  the  heat  to  be  rejected  ; 

the  compressor  mechanically  raises  the  temperature  without  the  addition  of  heat ; 

cooling  of  the  fluid  occurs  during  its  passage  through  the  expansion  valve. 
The  path  through  the  expansion  valve  is  one  of  constant  total  heat ;  otherwise,  the 

cycle  is  ideally  that  of  Clausius. 

Q  =  xl  —  (H—h},  q  =  XL,  W  =  XL  +  II—  h  —  xl,  for  vapor  wet  throughout  com- 
pression. 

The  vapor  may  be  wet,  dry,  or  superheated  at  the  beginning  of  compression. 
The  fluid  used  should  be  one  having  a  large  latent  heat  and  small  specific  heat.     NH3, 

S(>2,  and  CO*  are  those  principally  employed. 
Capacity  =  ice-melting  effect  in  tons  per  24  hours  =  ^  per  24  hours,   corrected  for 

superheating. 

Economy  =  ice-melting  effect  per  pound  of  coal  or  per  Ihp.-hr. 
Calculations  of  economy,  capacity,  and  dimensions  must  include  the  corrective  factor 

(1-0.0^). 

The  absorption  machine  replaces  the  compressor  by  the  absorber  and  the  generator, 
For  low  vaporizer  temperatures  it  is  theoretically  superior  to  the  compression 
apparatus.  The  absorption  apparatus  should  give  an  efficiency  equal  to  that  in  a 
non-condensing  engine-driven  compression  system  when  the  vaporizer  temperature 
is  5°,  and  to  that  in  a  condensing  engine  system  when  it  is  0°. 

Refrigeration  may  be  indirect,  by  direct  expansion  or  by  brine  circulation. 

In  ice  making  the  can  system  is  more  rapid  and  occupies  less  space,  while  costing  less, 
than  the  stationary  cell  or  plate  system.  Clear  ice  is  produced  by  using  distilled 
water  and  as  high  a  temperature  as  possible.  An  economical  compressor  engine 
is  unnecessary.  The  pressure  range  is  usually  from  30  to  205  Ib.  The  actual  ice 
production  is  about  one  half  the  "  ice-melting  capacity." 

The  A.  S.  3f.  E.  basis  for  rating  machines  is  at  temperatures  of  0°  and  90°  F. 

Usual  piston  displacements  are  from  6500  to  8700  cu.  in.  per  minute  per  ton  of  rated 
capacity  ;  engine  power  rates,  from  1  to  2  Ihp.  per  ton. 

Efficiency  from  engine  cylinder  to  cooling  room  =  0.0557  x  ice-melting  effect  per 
Ihp.-hr. 

Efficiency  from  coal  to  cooling  room  =  0.01015  x  ice-melting  effect  per  pound  of  coal 
(14,000  B.  t.  u.). 

Usual  efficiencies  from  coal  to  cooling  room,  with  vapor  machines,  range  from  0.100  to 
0.469,  the  average  in  good  tests  being  about  0.237  ;  say  23^  Ib.  of  ice-melting  effect 
per  pound  of  coal.  Absorption  machines  have  not  shown  efficiencies  quite  as  high  ; 
those  of  air  machines  are  extremely  low. 


422 


APPLIED  THERMODYNAMICS 


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APPLIED  THERMODYNAMICS 


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PROBLEMS  425 


PROBLEMS 

NOTE.  Our  knowledge  of  the  properties  of  some  of  the  vapors  used  in  refrigeration 
is  far  from  accurate.  Any  general  conclusions  drawn  from  the  results  of  the  problems 
are  therefore  to  be  regarded  with  caution.  (See  Art.  402.) 

1.  Plot  to  scale  to  PV  coordinates  a  Carnot  cycle  for  air  in  which  T=  80°  F., 
t  =  0°  F.,  and  the  extreme  range  of  specific  volumes  is  from  1  to  4.     Compare  its  area 
with  that  of  the  Joule  cycle  between  the  same  volume  and  pressure  limits. 

2.  In  a  Bell-Coleman  machine  working  between  atmospheric  and  73.5  Ib.  pressure, 
the  temperature  of  the  air  at  the  condenser  outlet  is  80C,  and  that  at  the  compressor 
inlet  is  0°.     Find  the  temperatures  after  expansion  and  after  compression,  the  curves 
following  the  law  PF1-35  =  c. 

3.  Find  the  coefficient  of  performance  for  a  Bell-Coleman  machine  with  pressures 
and  temperatures  as  given  above,  but  with  compression  in  two  stages  and  intercool- 
ing  to  80°.     (The  intermediate  pressure  stage  to  be  determined  as  in  Art.  211.)     Com- 
pare with  that  of  the  single-stage  apparatus.  * 

4.  Compare  the  consumption  of  water  for  cooling  in  jackets  and  condenser  and 
for  intercooling,  in  the  two  cases  suggested.     (See  Art.  234.) 

5.  The  machine  in  Problem  2  is  to  handle  10,000  cu.  ft.  of  free  air  at  32°  F.  per 
hour.     Find  the  sizes  of  the  double-acting  expansion  and  compression  cylinders  ideally 
necessary  at  100  r.  p.  m.  and  400  ft.  per  minute  piston  speed. 

6.  What  would  be  the  sizes  of  compressive  cylinders,  under  these  conditions,  if 
compression  were  in  two  stages  ? 

7.  Find  the  theoretical  cylinder  dimensions,  power  consumed,  coefficient  of  per- 
formance, and  cooling  water  consumption,  for  a  single-stage,  double-acting,  dense  air 
machine  at  60  r.  p.  m.,  300  ft.  per  minute  piston  speed,  the  pressures  being  65  and  225 
Ib.,  the  compressor  inlet  temperature  5°,  the  condenser  outlet  temperature  of  air  95°, 
and  the  circulating  water  rising  from  65°  to  80°.     The  apparatus  is  to  make  \  ton  of 
ice  per  hour  from  water  at  65°.     The  curves  follow  the  law  pvlM=  c. 

8.  Find  the  theoretical  coefficient  of  performance  of  a  sulphur  dioxide  machine 
working  between  temperatures  of  64.4°  and  5°  F.,  the  condition  at  the  beginning  of 
compression  being,  (a)  dry,  (ft)  60  per  cent  dry.    Also  (c)  if  the  substance  is  dry  at 
the  end  of  compression. 

9.  Check  all  values  in  Art.  632. 

10.  What  is  the  theoretical  ice- melting  capacity  of  the  machine  in  Problem  5  ? 

11.  Find  the  ice-melting  capacity  per  horse  power  hour  in  Problem  7. 

12.  Find  the  results  in  Problem  7  for  an  ammonia  machine  working  between  5°  and 
95°,  the  vapor  being  just  dry  at  the  end  of  adiabatic  compression.     How  do  the  coeffi- 
cients of  performance  in  the  two  cases  compare  with  those  of  the  corresponding  Carnot 
cycles  ? 

13.  What  is  the  loss  in  Problem  2,  if  a  brine  circulation  system  is  employed,  re- 
quiring that  the  temperature  at  the  compressor  inlet  be  —  25°  F.  ? 

*  The  formula  for  coefficient  of  performance  of  the  Joule  cycle,  given  in  Art.  622, 
will  be  found  not  to  apply  when  the  paths  are  not  adiabatic. 


426  APPLIED  THERMODYNAMICS 

14.  In  a  Kelvin  warming  machine,  the  temperature  limits  for  the  engine  are 
300°  F.  and  110°  F. ;  those  for  the  warming  cycle  are  150°  F.  and  60°  F.     Assume  that 
the  cycles  are  those  of  Carnot,  and  introduce  reasonable  efficiency  ratios,  determining 
the  probable  efficiency  (referred  to  power  only)  of  the  entire  apparatus. 

15.  Compare  the  coefficient  of  performance  in  Problem  12  with  those  in  which  the 
vapor  is  (a)  80  per  cent  dry,  (6)  dry,  as  it  leaves  the  vaporizer. 

16.  Find  the  coefficient  of  performance  in  Problem  15  (&)  if  the  compressive  path 
is  PF1-25  =  c.     (Compare  the  Pambour  cycle,  Art.  413.) 

17.  Compare  the  ratio  latent  heat at  &0  ftnd  64  4o  F    (An    632)    for 

specific  heat  of  liquid 
ammonia,  carbon  dioxide,  and  sulphur  dioxide.    Draw  inferences. 

18.  Plot  on  the  entropy  diagram  in  Problem  12  the  path  of  the  substance  through 
the  expansion  valve,  determining  five  points. 

19.  Find  the  temperature  at  the  generator  discharge  of  an  ammonia  absorption 
machine,  the  liquid  from  the  absorber  being  delivered  at  110°  F.  and  30  Ib.  pressure, 
and  the  pressure  of  vapor  leaving  the  generator  being  198  Ib. 

20.  An  ammonia  compression  apparatus  is  required  to  make  200  tons  of  ice  per 
24  hr.;  in  addition  it  must  cool  1,000,000  cu.  ft.  of  air  from  90°  to  40°  each  hour  by 
indirect  refrigeration.     Making  allowances  for  practical  imperfections,  find  the  tonnage 
rating,  cylinder  dimensions,  power   consumed,  cooling  water  consumption,  and   ice- 
melting  effect  per  Ihp.-hr.,  the  machine  being  double-acting,  70  r.  p.  m.,  560  ft.  per 
minute  piston  speed,  operating  between  33.67  and  198  Ib.  pressure  with  vapor  dry  at 
the  end  of  adiabatic  compression,  water  being  available  at  65°.     Estimate  whether  the 
exhaust  steam  from  the  engine  will  provide  sufficient  water  for  ice  making. 

21.  Make  an  estimate  of  the  production  of  ice  per  pound  of  coal  in  a  good  plant. 

22.  What  is  the  tonnage  rating  of  the  machine  in  Problem  20  on  the  A.S.M.E. 
basis? 

23.  Coal  containing  13,500  B.  t.  u.   per  pound   drives  a  simple   non-condensing 
engine  operating  an  ammonia  compression  apparatus.     The  ice-melting  effect  is  84  Ib. 
per  Ihp.-hr.  at  the  engine  cylinder.    The  coal  consumption  is  3  Ib.  per  Hip.-hr.*  and  the 
mechanical  efficiency  of  the  combined  engine  and  compressor  is  0.80.     Find  the  ice- 
melting  effect  per  pound  of  coal,  the  coefficient  of  performance,  the  efficiency  from  fuel 
to  engine  cylinder,  and  the  efficiency  from  fuel  to  refrigeration.     May  this  last  exceed 
unity  ? 

24.  An  absorption  apparatus  gives  an  ice-melting  effect  of  1.8  Ib.  per  pound  of 
dry  steam  at  27  Ib.  pressure  from  feed  water  at  55°  F.    Prove  that  this  performance 
may  be  excelled  by  a  compression  plant. 

25.  Find  a  relation  between  coefficient  of  performance  and  ice-melting  effect  per 
Ihp.-hr.  at  the  compressor  cylinder. 

26.  Find  the  tonnage  on  the  A.S.M.E.  standard  basis  of  a  12  x  30  inch  double- 
acting  compressor  at  60  r.  p.  m.,  using  (a)  ammonia,  (6)  carbon  dioxide. 

27.  Find  the  A.S.M.E.  tonnage  rating  for  an  ammonia  absorption  apparatus  work- 
ing between  30  and  182.83  Ib.  pressure  with  10,000  Ib.  of  dry  vapor  entering  the  gen- 
erator per  hour. 

28.  Check  all  derived  values  in  Art.  660  to  Art.  663. 


PROBLEMS 


427 


29.  Compare  the  coefficients  of  performance,  in  Art.  632  and  in  Problem  8,  with 
those  of  the  corresponding  Carnot  cycles. 

30.  Compute  the  value  of  JTin  Art.  658. 

31.  Compute  as  in  Art.  632  the  results  for  carbon  dioxide.     Why  might  not  ether 
be  included  in  a  similar  comparison  ? 

32.  Ether  at  52°  F.  is  compressed  adiabatically  to  232°  F.,  becoming  wholly  liquid. 
What  was  its  initial  condition  ?     (Fig.  315.) 

PRESSURE.     IBS.  PER.SQ.  IN. 

°      :  1*"       .    j»  - 

e>  o 


0.05 


0.15  0.20  0.25  0.30  0.35 

ENTROPY 

FIG.  315.  —Entropy  Chart  for  Ether. 


o.io 


33.  Discuss  variations  with  temperature  of  the  total  heat  of  ammonia,  sulphur 
dioxide,  carbon  dioxide.     (See  tables,  pp.  422-424.) 

34.  Plot  a  total-heat  entropy  diagram  for  carbon  dioxide. 

35.  Find  the  ratio  cubic  inches  of  piston  displacement  per  minute  for  the  A  s  M  E 

rated  tonnage 
temperature  limits,  with  vapor  dry  at  the  beginning  of  compression.     (Art.  659.) 

36.  Find  a  general  expression  for  the  coefficient  of  performance  in  the  Joule  cycle, 
the  paths  not  being  adiabatic. 

37.  Discuss  the  economy  and  general  desirability  of  using  the  exhaust  steam  from 
the  engine  driving  an  ammonia  compressor,  to  distill  in  vacuo  the  water  from  which  ice 
is  to  be  made. 


428 


APPLIED  THERMODYNAMICS 


XVI  i 


AMMONIA   ENTROPY  DIAGRAM 


429 


20         40 


Pressure,  Lbs.  per  Sq.  In. 

80        100        120       140        180       180 


220       240 


Entropy  above  32  °F.,  B.t.u; 
FIG.  316.  —  Entropy  Chart  for  Ammonia. 


INDEX 


(Referring  to  Art.  Nos.) 


Absolute  temperature,  152-156,  167. 
Absolute  zero,  44,  45,  156. 
Accumulator,  541. 

Adiabatic,  83,  100-105,  168,  173,  176,  325. 
Adiabatic  expansion  of  steam,  372,  373,  416, 

431,  432,  513,  515,  517-520. 
Acliabatics  of  vapors,  391-397. 
Afterburning,  325. 
Aftereooler,  208. 
Air,  Carnot  cycle  for,  249. 
liquid,  246,  609-610. 
specific  heat,  71,  72. 
Air    compressor,    193-212,    215,    216,  221- 

242,  540. 

Air  cooling  in  compressor,  199. 
Air  engine,  177,  180-192,  245,  254. 
Air  refrigerating  machine,  227,  612-620,  661. 
Air  supply,   boiler  furnace,   560,   563-567, 

573,  575. 
gas  engine,  309. 

Air  thermometer,  41,  42,  48,  49,  152. 
Air  transmission,  243,  245. 
Alcohol  engine,  279,  280,  341. 
Alcohol  thermometer,  7. 
Ammonia,  403,  606,  630-632,  644,  Table, 

p.  422,  Fig.  316. 
Ammonia    absorption    machine,    639-643, 

663. 

Analysis  of  producer  gas,  285. 
Andrews'  critical  temperature,  379,  380,  605, 

607. 

Anthracite  coal,  560. 
Apparent  ratio  of  expansion,  450. 
Apparent  specific  heat,  61. 
Atkinson  gas  engine,  276,  296,  297. 
Atmospheric  condenser,  664. 
Atomic  heat,  59. 
Automatic  engine,  507. 
Automobile  engine,  335,  340,  348. 
Auxiliaries,  gas  producer,  281. 
Avogadro's  principle,  40,  53,  56. 

Back  pressure,  448,  459. 

Barometric  condenser,  584. 

Barrel  calorimeter,  489. 

Bell-Coleman   refrigerating   machine,   613- 

620,  661. 

Bicycle,  motor,  340. 
Binary  vapor  engine,  483. 
Blast  furnace  gas,  276,  278,  329,  353. 
Blowing  engine,  179. 


Boiler,  566,  568-574. 

Boiler  efficiency,  569,  571-574, 

Boiler  horse-power,  570. 

Boiler  surface  efficiency,  574. 

Boulvin's  method,  455,  456. 

Boyle's  law,  38,  39,  84. 

Brake  horse  power,  555. 

Brauer's  method,  117. 

Brayton  cycle,  299,  300,  302,  304. 

Brine,  646,  657. 

Brine  circulation,  645-647. 

British  thermal  unit,  22. 

Bucket,  512,  527-530. 

Caloric  theory,  2,  131. 

Calorie,  23. 

Calorimeter,  488-494. 

Calorimetric  testing  of  steam  engine,  504, 

505,  511. 

Capacity,    air    compressor,    222-229,    230- 
237. 

air  engine,  182,  183,  188,  190. 

air  refrigerating  machine,  616,  618,  619. 

compound  steam  engine,  476—477. 

gas  engine,  277,  330. 

hot-air  engine,  248,  249,  275,  277. 

Otto-cycle  gas  engine,  293. 

steam  cycles,  418. 

steam  engine,  446,  447,  449. 

vapor  compressor,  633,  636,  637. 
Carbon   dioxide,    379,    402,    605-608,    611, 

630-632,  637,  Fig.  314,  Table,  p.  423. 
Carbureted  air,  279. 
Carburetor,  279,  282,  310,  336. 
Cardinal  property,  10,  76,  81,  88,  160,  162, 

169,  176,  370,  399. 
Carnot,  28,  130. 
Carnot  cycle,  128-143,  451. 

air  engine,  250. 

entropy  diagram,  159,  166. 

for  air,  249. 

for  steam,  406. 

reversed,  138,  612,  621. 
Carnot  function,  155. 
Cascade  system,  608. 
Centigrade  heat  unit,  23. 
Centigrade  thermometer,  8. 
Change  of  state.  15-18. 
Characteristic    equation,    10,    50,    84,    363, 

390,  401,  403,  404. 
Characteristic  surface,  84. 


431 


432 


INDEX 


Charles'  law,  41-49,  84. 

Chimney,  575. 

Circulation  in  steam  boiler,  569. 

Clapeyron's  equation,  368. 

Clausius  cycle,  408,  410,  447,  514,  544. 

Clearance,  188. 

air  compressor,  222,  223. 

gas  engine,  313,  324. 

steam  engine,  450,  451,  462. 

vapor  compressor,  616,  618. 
Clerk's  gas  engine,  300,  303-305. 
Closed  feed- water  heater,  581. 
Closed  hot-air  engine,  248,  275. 
Coal,  560,  578. 
Coal  gas,  276,  278,  329. 
Coefficient  of  performance,  621,  622,  628. 
Coil  calorimeter,  490. 
Combined  diagrams,  466,  469-473,  475. 
Combustion,  560,  563-567,  569,  573,  575. 
Complete  pressure  gas   engine   cycle,   300, 

303-305. 
Compound  steam  engine,  438,  459-483,  510, 

550. 

Compound  locomotive,  510. 
Compressed  air,  177-247. 

distributing  system,  212-221. 

refrigeration  by,  227. 

storage  system,  185,  245. 

transmission,  243-245. 
'uses,  177,  178. 
Compression,  in  air  compressor,  195—211. 

air  engine,  189,  191. 

Carnot  cycle,  132,  134. 

gas  engine,  276,  295,  297,  299,  312,  313, 
325,  348. 

steam  engine,  451,  462. 
Compressive  efficiency,  213. 
Compressor,   air,    193-212,   215,   216,   221- 
242,  540. 

vapor,  624-638,  642,  658,  660,  662,  664. 
Condensation  in  steam  cylinder,  428—443, 

448,  460. 

Condenser,  502,  584,  585,  617,  635;  664. 
Constant  dryness  curve,  369. 
Constant  heat  curve,  370,  398. 
Constant  volume  curve,  377. 
Constant  weight  curve,  365. 
Constrained  expansion,  124. 
Cooling  of  gas  engine  cylinder,  312,  314- 

318,  325. 

Cooling  tower,  585,  664. 
Cooling  water  in  refrigerating  plant,   617, 

635. 

Coordinate  diagrams,  81—127,  158. 
Criterion  of  reversibility,  139-141,  144-149. 
Critical  temperature,  379,  380,  605,  607. 
Cross-compound    steam   engine,    464,    470, 

472. 

Curtis  steam  turbine,  524,  531,  537. 
Cushion  air,  262,  264. 
Cushion  steam,  453,  457,  575. 
Cycle,  Carnot,  128-143,  451. 


Cycle,  external  work,  89. 

forms,  130. 

heat  engine,  129. 
Cycle,  heat  expended  in,  90. 

regenerative,  259. 

reversed,  90. 

reversible,   138-141,   147,   148,   152,   175, 

176. 
Cycles,  air : 

air  compressor,  194-211. 

air  engine,  180-183. 

air  refrigeration,  615. 

air  system,  218-221. 

Bell-Coleman,  615. 

Ericsson,  270. 

hot-air  engine,  256. 

Joule,  254,  255,  613,  622. 

Lorenz,  252. 

polytropic,  251. 

regenerative  air  engine,  259. 

Reitlinger,  253. 

Stirling,  264. 
Cycles,  gas,  276,  287-308. 

Brayton,  299. 

Clerk,  300,  303-305. 

complete  pressure,  300,  303-305. 

Diesel,  306,  307. 

Lenoir,  298,  300,  301,  304. 

Otto,  276,  287-297,  300,  309-329. 

two-stroke,  289-292,  329. 
Cycles,  refrigerative  : 

air  machine,  254,  255,  613,  615,  622. 

regenerative,  259,  610,  612. 

vapor  machine,  627. 
Cycles,  steam,  417,  418,  422-458. 

binary,  483. 

Clausius,  408-410,  417,  447,  514,  544. 

non-expansive,  412,  417,  423. 

Pambour,  413,  417. 

Rankine,  411,  417,  424,  429,  447. 

superheated,  414-418. 

turbine,  514. 

Cylinder  condensation,  428-443,  448,  460. 
Cylinder  efficiency,  212,  215,  216,  229. 
Cylinder  feed,  453,  475. 
Cylinder  ratios,  476,  477,  480. 
Cylinder  walls,  429,  431,  432,  504,  505. 

vapor  compressor,  637. 

Dalton's  law,  40. 

Davis'  method  for  determining  H,  360,  388. 

De   Laval   steam   turbine,    512,    524,    530, 

536.  * 

Dense  air  refrigerating  machine,  620. 
Desormes'  apparatus,  110. 
Diagram,  coordinate,  81-127,  158. 

entropy,  158-160,  164,  166,  169-171, 
174,  184,  218-221,  255,  266,  307,  347, 
367,  376-378,  398,  453-458,  475,  615, 
627. 

indicator,  437,  452,  454,  486-487,  500, 
501. 


INDEX 


433 


Diagram,  indicator,  gas  engine,  311. 

Mollier,  399,  516,  532. 

of  energy,  86—90. 

temperature-entropy,  158-160,  164,  166, 
169-171,  174,  184,  218-221,  255,  266, 
307,  347,  367,  376-378,  398,  453-458, 
475,  615,  627. 

total  heat-entropy,  399,  516,  532. 

total  heat-pressure,  399. 

velocity,  527-529,  534. 
Diagram  factor,  329,  446,  475,  633. 
Diesel  engine,  306,  307. 
Difference  of  specific  heats,  65,  67,  77,  165. 
Direct  expansion,  644. 
Disgregation  work,   3,    12,    15-17,   53,   56, 

64,  75,  76,  78,  80,  359,  360. 
Dissipation  of  energy,  176. 
Dissociation,  63,  318,  325. 
Distillation,  591-601. 
Distribution    of    work,    compound    steam 

cylinders,  464,  467,  468,  478. 
Double-acting  engine,  423. 
Draft,  560,  567,  575-577,  582. 
Drop,  181,  436,  447,  465,  467,  468,  479. 
Dry  compression,  629,  647. 
Dry  vacuum  pump,  237,  584. 
Dryness  curve,  369. 
Duplex  compressor,  239. 
Duty,  503. 

Economizer,  282,  582. 
Effects  of  heat,  12-17. 
Efficiency,  air  engine,  180,  185,  190,  192. 

boiler,  569,  571-574. 

boiler  furnace,  574. 

Brayton  cycle,  299. 

Carnot  cycle,  135,  136,  142,  166. 

Clausius  cycle,  409. 

compressed  air  system,  212—217. 

compressive,  213. 

Diesel  engine,  307. 

Ericsson  engine,  248,  249,  269-273. 

gas  engine,  334,  342-346. 

gas  producer,  284-286. 

heat  engine,  128,  142,  143,  149. 

injector,  588,  590. 

Joule  air  engine,  235. 

Lenoir  cycle,  298,  301. 

mechanical,  212,  214,  216,  342,  345,  487, 
503,  511,  546,  554-559. 

multiple-effect  evaporation,  599. 

non-expansive  cycle,  412. 

Otto  gas  engine,  295-297,  300. 

Pambour  cycle,  413,  417. 

plant,  503. 

Rankine  cycle,  411. 

refrigerating  machine,  621,  622,  634, 
642,  661-663. 

refrigerating  plant,  621,  622,  628. 

steam  engine  and  turbine,  546,  553. 

steam  turbine,  526,  529. 

Stirling  engine,  265,  267,  268. 


Efficiency,  superheated  cycles,  415. 

thermal,  342. 

transmissive,  212-216,  243,  244. 
Efficiency,  volumetric,  222-229.         ^ 
Ejector,  587. 
Electrical  ignition,  323. 
Electrical  resistance  pyrometer,  9. 
Electric  calorimeter,  494. 
Energy,   10,   12,  76-78,  81,   100,   109,   113, 

119-123,  359,  374,  375. 
Engine,  air,  177,  180-192,  245. 

binary  vapor,  483. 

blowing,  179. 

Clerk's,  300,  303-305. 

Diesel,  306-307. 

gas,   276,   277,   287-308,   312,   313,   324, 
325,  330,  348. 

heat,  128,  130,  132,  139,  142,  143. 

hot-air,  248-275,  277. 

internal  combustion,  248,  276,  277,  287- 
308. 

Joule,  235,  254. 

oil,  276,  279,  280,  299. 

rotary  steam,  177,  192. 

steam,  408^19,  422-511,  514,  544,  550. 

turbine,  239,  512-542,  552,  553,  556. 
Entropy,  10,  157-176. 

formulas,  169. 

gases,  169. 

physical  significance,  160. 

units,  171. 
Entropy  diagram,  158,  174. 

air  engine,  184. 

Bell-Coleman  machine,  615. 

Carnot  cycle,  159,  166. 

compressed  air  system,  218-221. 

Diesel  engine,  307. 

gas  engine,  347. 

Joule  engine,  255. 

specific  heats  of  gases.  164. 

steam,  398. 

steam  engine,  453-458,  475. 

steam  formation,  367,  376-378. 

Stirling  engine,  266. 

vapor  refrigeration,  627. 
Equation,   characteristic,    10,   50,   84,   363, 
390,  401,  403,  404. 

of  condition,   10,  50,  84,   363,  390,  401, 

403,  404. 

Equation  of  flow,  522. 
Equivalent    evaporation,    361,    367,    386, 

572. 

Ericsson  hot-air  engine,  270. 
Ether,   371,  372,  402,   483,   611,   631,   663, 

Fig.  315. 
Evaporation,  factor  of,  361,  367,  389,  572. 

in  vacuo,  591-601. 

latent  heat  of,  359,  360. 

rate  of,  569. 

Evaporative  condenser,  585. 
Evaporator,  593,  595,  600,  601. 
Exhaust  line,  gas  engine  diagram,  326. 


434 


INDEX 


Exhaust  steam  injector,  589. 
Exhaust  steam  turbine,  541. 
Expansion,  constrained,  124. 

direct,  644. 

Expansion,   free,  73,  75,  79,  124-127,  513, 
515,  517,  607,  610. 

latent  heat  of,  58,  107. 

regenerative,  610,  612. 

steam  cylinder,  428-447,  450,  473,  486, 
558. 

steam  turbine,  513,  515,  517. 
Expansion  curve,  gas  engine,  325. 
Explosion  waves,  319,  325. 
Exponential  equation,  391,  394,  395. 
External  work,  14,  15,  86-90,  95,  98,  121- 

123,  160,  359,  374,  375. 
Externally  fired  boiler,  568. 

Factor,  heat,  170. 

Factor  of  evaporation,  361,  367,  389,  572. 

Feed  pump,  586. 

Feed-water  heater,  580-582. 

Figure  of  merit,  286. 

Fire-tube  boiler,  568,  569. 

First  law  of  thermodynamics,   28-37,   79, 

128,  167,  505. 
Fixed  point,  6,  16,  18. 

Flame  propagation,  309,  310,  319,  320,  325. 
Flow,  equation  of,  522. 

in  nozzle,  521-523. 

in  orifice,  523. 
Fluid  friction,  326,  342. 
Forced  draft,  577. 

Formation  of  steam,  354-360,  366,  381,  386. 
Free  expansion,  73,  75,  79,   124-127,  513, 

515,  517,  607,  610. 
Freezing  mixtures,  15,  611. 
Friction,  fluid,  326,  342. 

in  Joule's  experiment,  76,  127. 

in  nozzles,  518-520,  523. 

in  steam  engine,  555-559. 

in  turbine  buckets,  527. 
Fuel  oil,  280. 
Fuels,  560,  561. 
Function,  Carnot's,  155. 

thermodynamic,  170. 
Furnace  efficiency,  574. 
Fusion,  602-604. 

Gas,  coal,  276,  278,  329. 

liquefaction  of,  605-610. 

natural,  276,  278,  329. 

oil,  279. 

perfect,  39,  50,  51,  53,  56,  74,  80,  607. 

permanent,  16,  63,  605. 

producer,  276-286,  312,  329. 

steam,  357,  390,  391. 

water,  278,  281,  329. 

Gas  engine,   276,   277,   287-308,   312,   313, 
324,  325,  330,  348. 

Clerk's,  300,  303-305. 
Gas  engine  design,  330-335. 


Gasoline,  279,  280. 

Gas  power,  276-353. 

Gas  producer,  276-286. 

Gas  producer  auxiliaries,  281. 

Gas  transmission,  276. 

Gas  turbine,  540. 

Gases,  kinetic  theory,  53-56,  80. 

Gay-Lussac's  law,  41-49. 

Goss  evaporator,  601. 

Governing,  air  compressor,  238. 

gas  engine,  336-338,  348,  349. 

steam  engine,  462,  478. 
Gram-calorie,  23. 
Gravity  return  drip  system,  583. 

H,  359,  360,  388. 

Heat    absorbed,    graphical    representation, 

106,  123,  167. 

Heat,  mechanical  theory,  2-5. 
Heat  balance,  346,  496. 
Heat  drop,  515-519. 

Heat  engine,  128,  130,  132,  139,  142,  143. 
Heater,  feed-water,  580-582. 
Heat  factor,  170. 
Heat  of  liquid,  359. 
Heat  unit,  20-23. 
Heat  weight,  170,  172. 
Heating  surface,  569,  574. 
High-speed  steam  engine,  434,  507. 
High  steam  pressure,  143,  444,  459,  462. 
Him,  32. 

Hirn's  analysis,  504,  505,  511. 
Hit-or-miss  governing,  349. 
Horse  power,  boiler,  570. 

brake,  555. 

Hot-air  engine,  248-275,  277. 
Hot-air  jacket,  439. 
Hot-tube  ignition,  322,  336,  337. 
Hydraulic  compressor,  241. 
Hydraulic  piston  compressor,  240. 
Hydrogen,  60,  609. 

in  producer  gas,  284,  285,  312. 
Hyperbolic  curve,  445,  450,  473,  486. 

Ice,  2,  85,  602-604. 
Ice  making,  652-657. 
Ice-melting  effect,  634. 
Ignition,  314-323,  325,  336,  337. 
Impulse  turbine,  524,  530-533,  536-538. 
Incomplete  expansion,   181,  436,  447,  465, 

467,  468,  479. 

Indicated  thermal  efficiency,  342. 
Indicator,  424,  484-485. 
Indicator  diagram,  437,  452,  454,  486-487, 

500-501. 
gas  engine,  311. 
Indirect  refrigeration,  648. 
Induced  draft,  577. 
Initial    condensation,    430,    433,    436,    437, 

442,  448,  460. 
formula  for,  437. 
Injector,  587-590. 


INDEX 


435 


Injector  condenser,  584. 

Injection  of  water,  195,  200. 

Intercooler,  206,  207. 

Intermediate  compound,  480. 

Internal  combustion  engine,  248,  276,  277, 

287-308. 
Internal   energy,    10.    12,    76-78,    81,    100, 

109,  113,  119-123,  359,  374,  375. 
Internal  work  of  vaporization,  359,  360. 
Internally  fired  boiler,  568,  569. 
Inversion,  373,  395,  401. 
Irreversibility,  11,  73-76,  78,  175,  176. 
Irreversible  process,  124-127,  160,  426,  513. 
Isentropic,  168,  176. 
Isodiabatic,  108,  112. 
Isodynamic,  83,  96,  120-122. 
Isodynamic,  vapor,  382. 
Isoenergic,  83,  96,  120-122,  382. 
Isometric,  83. 
Isopiestic,  83. 
Isothermal,  78,  83,  91-95,  122,  366. 

Jacket,  gas  engine,  352,  353. 

hot-air,  439. 

steam,  413,  438-441,  475,  482,  505. 

vapor  compressor,  635. 
Jet  condenser,  502,  584. 
Joule  air  engine,  254. 
Joule  apparatus,  2,  30. 
Joule  cycle,  254,  255,  613,  622. 
Joule  experiment,  73-80,  124-127,  156,  176. 
Joule's  law,  75-80,  109. 

Kelvin  scale  of  absolute  temperature,  153- 

156,  167. 

Kelvin  warming  machine,  623. 
Kerosene,  279,  280. 
Kinetic  theory  of  gases,  53-56,  80. 
Kirk  air  refrigerating  machine,  612. 
Knoblauch  and  Jakob,  384. 

Lagging,  439. 

Latent  heat,  of  expansion,  58,  107. 

of  fusion,  602-604. 

of  evaporation,  359,  360. 
Lenoir  cycle,  298,  300,  301,  304. 
Linde  apparatus,  246,  610. 
Line  of  inversion,  373. 
Liquefaction  of  gases,  605-610. 

of  steam  during  expansion,  372,  373,  431, 

432. 

Liquid  air,  246,  609,  610. 
Locomotive  boiler,  568. 

superheater,  443,  509,  554. 

tests,  497,  5 11-,  554. 

theory,  509. 

turbo-,  540. 

types,  509,  510. 
Loop,  steam,  583. 
Lorenz  cycle,  252. 
Losses  in  steam  boiler,  566. 

in  steam  turbine,  514. 


Mariotte's  law,  38,  39. 

Marine  boiler,  568. 

Marine  turbine,  540. 

Mathematical  thermodynamic  method,  400, 

401. 

Mayer,  29,  72. 
Mayer's  principle,  94. 

Mean  effective  pressure,  331,  446,  476,  486. 
Mean  specific  heat,  61,  164. 
Mechanical  draft,  576,  582. 
Mechanical  efficiency,  212,  214,  216,  487, 

503,  511,  546,  554-559. 
gas  engine,  342,  345. 
Mechanical  equivalent  of  heat,   2,   28-37, 

79,  505. 

Mechanical  theory  of  heat,  2-5. 
Mercurial  thermometer,  7. 
Metallic  pyrometer,  9. 
Mixtures,  20,  21,  25. 
freezing,  15,  611. 
in  gas  engine,  309,  310,  348. 
Molecular  heat,  59. 
Mollier  diagram,  399,  516,  532. 
Mond  gas,  278,  283. 
Motor-bicycle,  340. 
Multiple-effect  evaporation,  594-601. 
Multiple  expansion,  438,  459-483,  550. 
Multi-stage  air  compression,  205-211,  221, 

226,  232,  234,  235,  239. 
Multi-stage  vapor  compression,  629. 

n,  91,  97,  115-118,  164. 
Natural  gas,  276,  278,  329. 
Negative  specific  heat,  115,  371. 
Negative  work,  87,  89,  99. 
Neutrals,  319,  320. 
Newhall  evaporator,  593. 
Non-expansive  cycle,  412,  423. 
Nozzle,  512-515,  518-523,  525. 

Oil  engine,  276,  279,  280,  299,  306,  307. 

Oil  fuel,  280. 

Oil  gas,  279. 

Open  feed-water  heater,  581. 

Opposed  beam  engine,  464. 

Optical  pyrometer,  9. 

Orifice,  523. 

Otto  cycle,  276,  287-297,  300,  309-329. 

Overload  capacity,  gas  engine,  330,  333. 

steam  engine,  447. 
Oxygen,  606,  608,  609. 

Pambour  cycle,  413. 

Parsons  turbine,  524,  533,  539,  556. 

Path,  83,  85,  88,  97-99,  111-118. 

Paths  of  vapors,  392-399. 

Pelton  bucket,  529. 

Perfect  gas,  39,  50,  51,  53,  56,  74,  80,  607. 

Permanent  gas,  16,  63,  605. 

Pictet  apparatus,  608. 

Pictet  fluid,  631. 

Piston  speeds,  gas  engine,  320. 


436 


INDEX 


Plant  efficiency,  503. 
Pneumatic  tools,  178. 
Poly  tropic  cycle,  251. 
Polytropic  paths,  97-99,  111-118,  125,  161, 

164,  165. 
Porous   plug  experiment,   73-80,    124-127, 

156,  176. 

Power  plant,  steam,  407,  408,  560-590. 
Preheater,  186,  187. 

Pressure,  high  steam,  143,  444,  459,  462. 
Pressure-temperature    relation,    355,    358, 

362,  368. 

Pressure  turbine,  524,  533-535,  539. 
Problems,  pages  10,  17-18,  28,  38,  60-62,  69, 

75-76,  88-89,  127-128,  144-145,  195- 

198,  252-255,  312-316,  347-349,  358- 

359,    378-379,    392-393,    425-427. 
Producer,  276-286. 
Producer  gas,  276-286,  312,  329. 
Propagation  of  flame,  309,  310,  319,  320, 

325. 
Properties  of  steam,  360,  367,  376,  405,  420, 

421. 

Pulsometer,  506. 
Pump,  feed,  586. 
pulsometer,  506. 
turbo-,  540. 

vacuum,  236,  237,  584,  591 
Pyrometer,  9. 

Quadruple  expansion  engine,  461,  476,  550. 

R,  51,  52,  65,  66,  68,  70. 

Rankine,  151. 

Rankine  cycle,  411,  424,  429,  447. 

Rankine's  theorem,  106,  157,  158,  167. 

Rateau  turbine,  524,  531,  538,  541. 

Rate  of  combustion,  560,  569. 

of  evaporation,  569. 

of  flame  propagation,  309,  310,  319,  320, 

325. 
Ratio  of  expansion,  433,  436,  446,  447,  459. 

compound  engines,  476. 

real  and  apparent,  450. 
Ratio  of  specific  heats,  69,  70. 
Reaction  turbine,  524,  533-535,  539. 
Real  ratio  of  expansion,  450. 
Real  specific  heat,  61,  78. 
Reaumur  thermometer,  8. 
Receiver  compound  engine,  464,  466-473. 
Receiver  pressure,  air  compressor,  211. 
Recuperator,  281. 
Reevaporation,  431,  445,  448,  460. 
Reeves'  method,  457. 

Refrigerating  machine,  612,  616,  618-620, 
629,  633,  636,  637,  647,  658-660,  663, 
664. 
Refrigeration,  611-664. 

applications  of,  649-657. 

compressed  air,  227,  247. 

vapors  used,  400-405. 
Regenerative  expansion,  610,  612. 


Regenerator,  246,  257-259,  281,  541,  610. 
Regnault,  43,  46,  49. 
Regnault's  law,  63. 
Regulation,  air  compressor,  238. 

gas  engine,  336-338,  348,  349. 

steam  engine,  462,  478. 
Reheating,  481. 
Reitlinger  cycle,  253. 
Representation  of  heat  absorbed,  106,  123, 

167. 
Reversibility,  139-141,  144-149. 

cycle,  138-141,  147,  148,  152,  175,  176. 

path,  125,  126,  162,  168,  175,  176. 
Rotary  steam  engine,  177,  192. 

Saturated  steam,  356,  358-382. 

Saturated  vapor,  356. 

Saturation  curve,  365. 

Scales,  thermometric,  8. 

Scavenging,  312,  327,  339. 

Second  law  of  thermodynamics,    138-142, 

144-156. 

Siphon  condenser,  584. 
Small  calorie,  23. 
Soft  coal,  560,  578. 
Solution,  15,  604. 
Sommeiller  compressor,  240. 
Specific  heat,  20,  21,  24-27,  57,  58. 

air,  71,  72. 

apparent,  61. 

difference,  65,  67,  77,  165. 

entropy  diagram,  164. 

gases,  57-72. 

mean,  61,  164. 

negative,  115,  371. 

polytropics,  112,  115,  164. 

ratio,  68,  70. 

real,  61,  78. 

saturated  vapor,  401. 

superheated  steam,  383-385,  387,  388. 

volumetric,  60,  67. 

water,  24,  26,  359. 

Specific  volume  of  steam,  360,  363,  368. 
Starting  gas  engines,  351. 
Steam,  formation,  354-359,  366,  381. 

pressure-temperature  relation,  355,  358, 
362,  368. 

saturated,  356,  358-382. 

superheated,    355,    358,    365,    366,    380, 

383-397. 
Steam  adiabatic,  372,  373,  431,  432,  513, 

515,  517-520. 

Steam  boiler,  566,  568-574. 
Steam  consumption,  546,  553. 

from  indicator  diagram,   437,   487,   500, 

501. 
Steam  cycles,  408-412,  414-418,  422-458, 

483,  514,  544. 
Steam  engine,  419,  422-511,  550. 

cycle,   408-412,   414-418,   422-458,   483, 
514,  544. 

description,  422. 


INDEX 


437 


Steam  engine,    entropy   diagram,   453-458, 
475. 

governing,  462,  478. 
Steam-ether  engine,  483. 
Steam  gas,  357,  390,  391. 
Steam  jacket,  413,  438-441,  475,  482,  505. 
Steam  loop,  583. 

Steam  power  plant,  407,  508,  560-590. 
Steam  rate,  546,  553. 
Steam  refrigeration,  631,  632,  638,  643. 
Steam  table,  360,  367,  376,  405,  420,  421. 
Steam  turbine,  512-542,  552,  553,  556. 
Steam,  wet,  364,  367. 
Still,  591. 

Stirling  hot-air  engine,  260-268. 
Stoker,  578. 

Storage,  compressed  air,  185',  245. 
Straight-line  compressor,  239. 
Stumpf  turbine,  536. 
Sublimation,  17. 
Suction  producer,  281,  282. 
Suction  stroke,  gas  engine,  328. 
Sulphur  dioxide,  404,  483,  606,  608,  611, 

631,  632,  Table,  p.  424. 
Superheat,  locomotives,  443,  509,  554. 

refrigeration,  629,  633,  636,  647. 

turbines,  517,  552,  553. 

use  of,  438,  442-444,  482,  551-553,  579. 
Superheated  adiabatic,  416. 
Superheated  steam,  355,  358,  365,  366,  380, 
383-397. 

cycles,  414-418. 

table,  421. 

Superheated  vapor,  356. 
Superheaters,  579. 
Superheating  calorimeter,  491. 
Surface  condenser,  502,  584. 
Surface-condensing  calorimeter,  490. 
Synopses,  pp.  10,  17,  27-28,  37-38,  60,  69, 
76,  87-88,  125-127,  143-144,  193-195, 
249-252,   309-312,    346-347,    377-378, 
391-392,  420^*21. 

Table,  steam,  360,  367,  376,  405,  420,  421. 
Tandem-compound,    464,    466,    467,    469, 

471. 

Tank  calorimeter,  489. 
Temperature,  6,  19-21. 

absolute,  152-156,  167. 

gas  engine  cylinder,  312,  314-318. 

inversion,  373,  395,  401. 

measurement,  6—9. 
Testing  hot  air  engines,  274. 
Tests,  locomotive,  497,  511,  554. 

refrigerating  machine,  660-663. 

steam  boiler,  572. 

steam    engine,    484-505,    543-551,    553, 
555-559. 

steam  turbine,  543-545,  552,  553,  555,  556. 
Thermal  capacity,  57,  58. 
Thermal  efficiency,  342. 
Thermal  line,  83. 


Thermochemistry,  4,  40,  53,  56,  59. 
Thermodynamic  function,  170. 
Thermodynamic  surface,  84. 
Thermo-electric  pyrometer,  9. 
Thermometer,  7,  8. 

air,  41,  42,  48,  49,  152. 
Thermometric  scales,  8. 
Thermometry,  6-9. 
Theta-phi  diagram,  170. 
Thomas'  experiments,  385. 
Throttling,  388,  425,  426. 
Throttling  calorimeter,  491. 
Throttling  engine,  427,  507. 
Throttling,  gas  engine,  326,  348. 
Thrust  in  turbines,  528. 
Time  of  ignition,  321. 
Tonnage  rating,  658,  659. 
Total  heat-entropy  diagram,  399. 
Total  heat-pressure  diagram,  399. 
Total  heat,  saturated  steam,  359,  360,  388. 

superheated  steam,  386. 
Tower,  cooling,  585,  664. 
Transmission,  air,  243-245. 

gas,  276. 
Transmissive     efficiency,    212,     216,     243, 

244. 

Triple-expansion  engine,  461,  476,  480,  549. 
Tubes  in  boilers,  569. 
Turbine,  gas,  540. 

steam,  512-542,  552,  553,  556. 
Turbo-compressor,  239,  540. 
Turbo-locomotive,  540. 
Turbo-pump,  540. 

Two-cycle  gas  engine,  289-292,  329,  339. 
Two   specific   heats   of   gases,    57,   58,    62, 

64-72,  107,  165. 
Types,  air  compressor,  238-242. 

gas  engine,  336-341,  350-351. 

locomotive,  509,  510. 

multiple-expansion  engine,  461. 

steam  engine,  507. 

vapor  compressor,  664. 

Vacuum,  footnote,  p.  358. 
Vacuum  distillation,  591—601. 
Vacuum  pump,  236,  237,  584,  591. 
Valves,  air  compressor,  242. 

gas  engine,  310,  326,  350. 

steam  engine,  452,  468,  507. 
Vapor,  paths,  392-399. 

specific  heat,  401. 
Vapor  adiabatic,  391-397. 
Vapor  compressor,  624-638,  642,  658,  660, 

662,  664. 

Vapor  refrigeration,  624-643,  647,  662. 
Vaporization,  internal  work,  359,  360. 

latent  heat,  359,  360. 
Vaporizer,  279,  282,  310,  336,  612,  626. 
Vapors,  16,  17,  354-421. 

for  refrigeration,  400-405,  630-632. 

saturated,  356. 

superheated,  356. 


438 


INDEX 


Velocity  diagram,  527,  529,  534. 
Velocity  turbine,  524,  530-533,  536-538. 
Velocity  work,  127,  176,  512,  525,  518. 
Vibration  work,  3,  12,  13,  54-56. 
Volume  curves  377. 
Volumetric  efficiency,  222-229. 
Volumetric  specific  heat,  66,  67. 
Volume,  vapor,  360,  363,  368,  369^401. 

Walls,  gas  engine  cylinder,  312,  317, 325, 347. 

steam  cylinder,  429,  431,  432,  504,  505. 

vapor  compressor,  637. 
Warming  machine,  Kelvin,  623. 
Water,  air  compressor,  195-200,  206,  207. 

in  refrigerating  plant,  617,  635. 

specific  heat,  24,  26,  359. 
Water  gas,  278,  281,  329. 
Water  jacket,  air  compressor,  201,  204 

gas  engine,  312,  317,  325,  352,  353 
Water  supply,  evaporator,  600. 
Water-tube  boiler,  568,  569. 
Watt's  diagram,  86-90. 
Watt's  law,  359. 
Waves,  explosion,  319,  325. 


Westinghouse-Parsons  turbine,  539. 

Wet  compression,  629,  647. 

Wet  steam,  364,  367. 

Willans'  line,  556. 

Wiredrawing,  425,426,442, 448, 451, 474, 518. 

Woolf  engine,  463. 

Work,  Clausius  cycle,  410. 

compression,  210. 

external,  14,  15,  86-90,  95,  98,  121-123, 
160,  359,  374,  375. 

negative,  87,  89,  99. 

superheated  adiabatic,  416. 

vapor  adiabatic,  396. 

velocity,  127,  176,  512,  515,  518. 

vibration,  3,  12,  13,  54-56. 
Wormell's  theorem,  36. 

y,  57,  58,    62,  64-72,  101,  102,  105,   107, 

110,  165. 
Yaryan  evaporator,  595-600. 

Zero,  absolute,  44,  45,  156. 

of  entropy,  171. 
Zero  line,  373. 


A   LIST   OF   BOOKS 

— ON— 


Steam  and  Steam  Engineering, 


AUCHINCLOSS,  W.  S.  The  Practical  Application  of  the  Slide 
Valve  and  Link  Motion  to  Stationary,  Portable,  Locomotive, 
and  Marine  Engines,  with  new  and  simple  methods  for  propor- 
tioning the  parts.  Fifteenth  Edition,  revised.  52  Illustrations.  8vo. 
Cloth.  144  pp $2.00 

BACON,  F.  W.  A  Treatise  on  the  Richards  Steam  Engine 
Indicator,  with  directions  for  its  use.  By  Charles  T.  Porter.  Re- 
vised, with  notes  and  large  additions  as  developed  by  American  prac- 
tice; with  an  appendix  containing  useful  formulae  and  rules  for  engi- 
neers. Fourth  Edition.  Illustrated.  16mo.  Cloth.  180  pp 1.00 

BARRUS,  G.  H.  Boiler  Tests:  Embracing  the  results  of  one 
hundred  and  thirty-seven  evaporative  tests,  made  on  seventy-one  boil- 
ers, conducted  by  the  author.  Illustrated.  Svo.  Cloth.  252  pp. 3. 00 

Engine  Tests :    Embracing  the  results  of  over  one  hundred  feed- 

"  water  tests  and  other  investigations  of  various  kinds  of  steam-engines, 
conducted  by  the  author.  With  numerous  figures,  tables,  and  dia- 
grams. Svo.  Cloth.  338  pp 4.00 

The  above  two  purchased  together 6 .00 

BEAUMONT,  W.  W.  Practical  Treatise  on  the  Steam  Engine 
Indicator,  and  Indicator  Diagrams.  With  notes  on  engine  per- 
formances, expansion  of  steam,  behavior  of  steam  in  steam  engine  cylin- 
ders, and  on  gas-  and  oil-engine  diagrams.  Second  Edition,  revised  and 
enlarged.  128  Illustrations.  Svo.  Cloth.  261  pp Net,  2 .50 

BEGTRUP,  J.  The  Slide  Valve  and  its  Functions.  With  special  refer- 
ence to  modern  practice  in  the  United  States.  Second  Edition.  Illus- 
trated. Svo.  Cloth.  145  pp 2.00 


2  D.  VAN  NOSTRAND  COMPANY'S 

BERTIN,  L.  E.  Marine  Boilers:  their  Construction  and  Work- 
ing, dealing  more  especially  with  tubulous  boilers.  Translated  by 
Leslie  S.  Robertson.  Preface  by  Sir  William  White.  Second  Edition, 
revised  and  enlarged  350  Illustrations.  8vo.  Cloth.  694  pp .  Net,  5 . 00 

BOOTH,  W.  H.  Superheat,  Superheating,  and  their  Control. 
Illustrated.  8vo.  Cloth.  170  pp Net,  1 .50 

Water  Softening  and  Treatment,  condensing  plant,  feed  pumps, 

and  heaters  for  steam  users  and  manufacturers.     Illustrated.      8vo. 
Cloth Net,  2.50 

CARPENTER,  R.  C.,  and  DIEDERICHS,  H.  Internal  Combus- 
tion Engines;  Their  Theory,  Construction,  and  Operation. 
Illustrated.  8vo.  Cloth.  611  pp ; Net,  5.00 

CHRISTIE,  W.  W.  Boiler-waters,  Scale,  Corrosion,  Foaming. 
77  Illustrations.  8vo.  Cloth.  242  pp Net,  3 . 00 

Chimney  Design  and  Theory.  A  book  for  engineers  and  archi- 
tects. Revised  and  enlarged.  Illustrated.  8vo.  Cloth.  200  pp. 3. 00 

Furnace   Draft ;     its  Production  by  Mechanical  Methods. 

Second  Edition,  revised.     Illustrated.     16mo.     Boards.     80  pp.     (Van 
Nostrand   Science  Series  No.   123.) 50  cents 

CLARK,  D.  K.  Fuel:  its  Combustion  and  Economy. 
Comprising  an  abridgment  of  "A  Treatise  on  the  Combustion  of  Coal," 
by  C.  W.  Williams.  With  extensive  additions  on  recent  practice  in  the 
combustion  and  economy  of  fuel,  coal,  coke,  wood,  peat,  petroleum, 
etc.  Fourth  Edition.  144  Illustrations.  12mo.  Cloth.  366  pp..  1.50 

DAY,  C.  Indicator  Diagrams  and  Engine  and  Boiler  Testing. 
Third  Edition.  125  Illustrations.  12mo.  Cloth.  220  pp 2.00 

DRAPER,  C.  H.  Heat  and  the  Principles  of  Thermodynamics. 
Third  Edition.  133  Illustrations.  12mo.  Cloth.  363  pp 1 . 50 

GOODEVE,  T.  M.  A  Text-book  on  the  Steam  Engine.  With  a 
supplement  on  gas  engines.  Twelfth  Edition,  enlarged.  143  Illustra- 
tions. 12mo  Cloth 2 . 00 

GOULD,  E.  S.  The  Arithmetic  of  the  Steam  Engine,  illus- 
trated. 12mo.  Cloth.  80  pp 1 .00 

HAEDER,  H.  A  Handbook  on  the  Steam  Engine.  With 
especial  reference  to  small  and  medium-sized  engines.  For  the  use 
of  engine  makers,  mechanical  draughtsmen,  engineering  students,  and 
users  of  steam  power.  Translated  from  the  German,  with  consid- 
erable additions  and  alterations,  by  H.  H.  P.  Powles.  Third  English 
Edition,  revised.  Illustrated.  8vo.  Cloth.  465  pp 3.00 


BOOKS  ON  STEAM  AND  ALLIED  SUBJECTS.  3 

HALSEY,  F.  A.  Slide  Valve  Gears.  An  explanation  of  the 
action  and  construction  of  plain  and  cut-off  slide  valves.  Eleventh 
Edition,  revised  and  enlarged.  109  Illustrations.  12mo.  Cloth.  211 
pp 1 .50 

HAUSBRAND,  E.  Drying  by  Means  of  Air  and  Steam.  With 
explanations,  formulas,  and  tables,  for  use  in  practice.  Translated 
from  the  German  by  A.  C.  Wright.  Illustrated.  12mo  Cloth. 
70  pp Net,  2 .00 

-  Evaporating,  Condensing,  and  Cooling  Apparatus.  Ex- 
planations, formulae,  and  tables  for  use  in  practice.  Translated  from 
the  second  revised  German  edition  by  A.  C.  Wright.  With  numerous 
figures,  tables,  and  diagrams.  8vo.  Cloth.  423  pp Net,  5.00 

HECK,  R.  C.  H.      The  Steam  Engine  and  Other  Steam  Motors, 

A  text-book  for  engineering  colleges  and  a  treatise  for  engineers.  In 
Two  Volumes.  8vo,  Cloth. 

Vol.  I.    The  Thermodynamics  and  the  Mechanics  of  the  Engine. 
187  Illustrations.     400  pp Net,  3.50 

Vol.   II.   Form,   Construction,   and  Working  of  the  Engine : 

The  Steam  Turbine.      698  Illustrations.      686  pp Net,  5 .00 

Abridged  edition  of  above  two  volumes In  Press 

HUTTON,  W.  S.  Steam  Boiler  Construction.  A  practical  hand- 
book for  engineers,  boiler  makers,  and  steam  users,  containing  a  large 
collection  of  rules  and  data  relating  to  recent  practice  in  the  design, 
construction,  and  working  of  all  kinds  of  stationary,  locomotive,  and 
marine  steam  boilers.  Fourth  Edition,  carefully  revised  and  enlarged. 
540  Illustrations.  8vo.  Cloth.  675  pp 6 .00 

The  Practical  Engineer's  Handbook.    Comprising  a  treatise  on 

modern  engines  and  boilers,  marine,  locomotive,  and  stationary.  Sixth 
Edition,  revised  and  enlarged.  423  Illustrations.  8vo.  Cloth.  556 
pp 7.00 

JAMIESON,  A.      A  Text-book  on  Steam  and  Steam  Engines, 

including  turbines  and  boilers.  Specially  arranged  for  the  use  of  engi- 
neers qualifying  for  the  Institution  of  Civil  Engineers,  the  diplomas 
and  degrees  of  technical  colleges  and  universities,  advanced  science 
certificates  of  British  and  colonial  Boards  of  Education,  and  honours 
certificates  of  the  City  and  Guilds  of  London  Institute,  in  mechanical 
engineering,  and  engineers  generally.  Fifteenth  Edition,  revised. 
Illustrated.  8vo.  Cloth.  842  pp ! 3 .00 

Elementary  Manual    on    Steam   and  the   Steam   Engine. 

Specially  arranged  for  the  use  of  first  year  science  and  art,  City  and 
Guilds  of  London  Institute,  and  other  elementary  engineering  students. 
Tenth  Edition, .revised.  12mo.  Cloth 1 . 50 


4  D.  VAN  NOSTRAND  COMPANY'S 

KENNEDY,    R.     Modern   Engines   and   Power   Generators.     A 

practical  work  on  prime  movers  and  the  transmission  of  power.     With 
tables,    figures,    and    full-page    engravings.     Six    volumes.    Illustrated. 

8yo.     Cloth 15.00 

Single  volumes,  each 3.00 

KERSHAW,    J.    B.    C.       Fuel,    Water,    and    Gas    Analysis, 

for  steam  users.     With  50  Illustrations.    _8vo.     Cloth.     177  pp.Net,  2.50 

KLEIN,  J.  F.  Design  of  a  High  Speed  Steam  Engine.  With 
notes,  diagrams,  formulas,  and  tables.  Second  Edition,  revised  and 
enlarged.  140  Illustrations.  8vo.  Cloth.  257  pp Net,  5 .00 

KLEINHANS,  F.  B.  Boiler  Construction.  A  practical  explanation 
of  the  best  modern  methods  of  boiler  construction,  from  the  laying  out 
of  sheets  to  the  completed  boiler.  With  diagrams  and  full-page  engrav- 
ings. 8vo.  Cloth.  421  pp 3.00 

KOESTER,  F.  Steam  Electric  Power  Plants  and  their  Con- 
struction. A  practical  treatise  on  the  design  of  central  light  and 
power  stations  and  their  economical  construction  and  operation.  340 
Illustrations.  4to.  Cloth.  473  pp Net,  5 .00 

LEASK,  A.  R.  Triple  and  Quadruple  Expansion  Engines  and 
Boilers  and  their  Management.  Fourth  Edition,  revised  and 
enlarged.  74  Illustrations.  12mo.  Cloth.  306  pp 2.00 

LEWES,  V.  B.  Liquid  and  Gaseous  Fuels  and  the  Part  They 
Play  in  Modern  Power  Production.  Illustrated.  8vo.  Cloth. 
348  pp.  (Van  Nostrand's  Westminster  Series.) Net,  2.00 

LUCKE,  C.  E.  Power,  Cost,  and  Plant  Designs  and  Construc- 
tion. 2  Vols In  Press 

PICKWORTH,  C.  N.  The  Indicator  Handbook.  A  practical  man- 
ual for  engineers.  In  Two  Parts.  12mo.  Cloth. 

Part  I.  The  Indicator:  Its  Construction  and  Application.  Third 
Edition.  81  Illustrations.  130  pp 1 . 50 

Part  II.  The  Indicator  Diagram:  its  Analysis  and  Calculation. 
Third  Edition.  148  Illustrations.  134  pp 1 . 50 

PRAY,  T.,  Jr.  Steam  Tables  and  Engine  Constant.  Compiled 
from  Regnault,  Rankine,  and  Dixon  directly,  making  use  of  the  exact 
records.  8vo.  Cloth.  126  pp 2 .00 

RANKINE,  W.  J.  M.  The  Steam  Engine  and  Other  Prime 
Movers.  With  diagram  of  the  mechanical  properties  of  steam.  Fold- 
ing-plates, numerous  tables,  and  illustrations.  Fifteenth  Edition,  thor- 
oughly revised  by  W.  J.  Millar.  8vo.  Cloth 5 . 00 

Useful  Rules  and  Tables  Relating  to  Mensuration,  Engi- 
neering, Structures,  and  Machines.  With  tables,  tests,  and 
formulae  for  the  use  of  electrical  engineers.  By  Andrew  Jamieson. 
Seventh  Edition,  thoroughly  revised  by  W.  J.  Millar.  Illustrated.  12mo. 
Cloth.  482pp. 4.00 


BOOKS  ON  STEAM  AND  ALLIED  SUBJECTS.  5 

RATEAU,  A.  Experimental  Researches  on  the  Flow  of  Steam 
Through  Nozzles  and  Orifices.  Translated  by  H.  B.  Brydon. 
Illustrated.  12mo.  Cloth.  82  pp Net,  1 . 50 

REED'S  Marine  Boilers.  A  treatise  on  the  causes  and  prevention 
of  their  priming,  with  remarks  on  their  general  management.  Third 
Edition,  rewritten  and  enlarged.  79  Illustrations.  12mo.  Cloth. 
264  pp Net,  2 .00 

RICHARDSON,  J.  The  Modern  Steam  Engine.  Theory,  Design, 
Construction,  Use.  A  practical  treatise.  300  Illustrations.  8vo. 
Cloth.  384  pp Net,  3.50 

ROBERTSON,  L.  S.  Water  Tube  Boilers.  Based  on  a  short  course 
of  lectures  delivered  at  University  College,  London.  With  171  Illus 
trations.  8vo.  Cloth.  228  pp 3 .00 

ROSE,  J.  Key  to  Engines  and  Engine  Running.  A  practical 
treatise  upon  the  management  of  steam  engines  and  boilers,  for  the 
use  of  those  who  desire  to  pass  an  examination  to  take  charge  of  an 
engine  or  boiler.  With  numerous  illustrations,  and  instructions  upon 
engineers'  calculations,  indicator  diagrams,  engine  adjustments,  and 
other  valuable  information  necessary  for  engineers  and  firemen.  Third 
Edition.  12mo.  Cloth.  410  pp 2 . 50 

ROSSITER,  J.  T.  Steam  Engines.  Illustrated.  8vo.  Cloth.  (Van 
Nostrand's  Westminster  Series.) In  Press 

ROWAN,  F.  J.  The  Practical  Physics  of  the  Modem  Steam 
Boiler.  With  an  introduction  by  R.  H.  Thurston.  With  314 
Illustrations.  8vo.  Cloth.  683  pp 7 . 50 

SCHUMANN,    F.      A    Manual    of    Heating    and    Ventilation 

in  their  practical  application,  for  the  use  of  engineers  and  architects. 
Embracing  tables  and  formulae  for  dimensions  of  heating,  flow  and 
return  pipes  for  steam  and  hot  water  boilers,  flues,  etc.  Fourth  Edition, 
revised  and  enlarged.  Illustrated.  12mo.  Leather.  100  pp 1 .50 

SCRIBNER,  J.  M.  Engineers1  and  Mechanics'  Companion. 
Twenty-first  Edition.  Illustrated.  16mo.  Morocco.  264  pp 1.50 

SEATON,  A.  E.  A  Manual  of  Marine  Engineering.  Comprising 
the  design,  construction,  and  working  of  marine  machinery.  With 
numerous  tables  and  illustrations  reduced  from  working  drawings. 
Sixteenth  Edition,  revised  and  enlarged.  8vo.  Cloth.  735  pp 6.00 

SEATON,  A.  E.,  and  ROUNTHWAITE,  H.  M.  A  Pocket-book  of 
Marine  Engineering  Rules  and  Tables.  For  the  use  of 
marine  engineers  and  naval  architects,  designers,  draughtsmen,  super- 
intendents, and  all  engaged  in  the  design  and  construction  of  marine 
machinery,  naval  and  mercantile.  Ninth  Edition,  revised  and  enlarged. 
Illustrated.  12mo.  Leather.  560  pp 3 .00 


6  BOOKS  ON  STEAM  AND  ALLIED  SUBJECTS. 

SEXTON,  A.  H.  Fuel  and  Refractory  Materials.  Illustrated.  8vo. 
Cloth 2.00 

SHOCK,  W.  H.  Steam  Boilers :  their  design,  construction,  and  man- 
agement. 150  Illustrations  and  34  Full-page  Plates.  4to.  Half 
morocco.  475  pp 15 . 00 

SOTHERN,  J.  W.  The  Marine  Steam  Turbine.  A  practical  de- 
scription of  the  Parsons  Marine  Turbine  as  now  constructed,  fitted, 
and  run.  Second  Edition,  revised  and  enlarged.  Illustrated.  8vo. 
Cloth.  173  pp Net,  2 .50 

STILLMAN,  P.  Steam  Engine  Indicator  and  the  improved  manom- 
eter steam  and  vacuum  gauges:  their  utility  and  application.  A'eiy 
Edition.  Illustrated.  16mo.  Flexible  cloth.  96  pp 1 .00 

STODOLA,  A.  Steam  Turbines.  With  an  appendix  on  gas  turbines, 
and  the  future  of  heat  engines.  Authorized  translation  by  Louis  C. 
Loewenstein.  With  241  cuts  and  3  lithographed  tables.  Second  Revised 
Edition.  8vo.  Cloth.  510  pp Net,  5 .00 

TONGE,  JAMES.  Coal.  Illustrated.  8vo.  Cloth.  283  pp.  (Van  Nos- 
trand's  Westminster  Series.) Net,  2.00 

TRINKS,  W.,  and  HOUSUM,  C.  Shaft  Governors.  27  Illustra- 
tions. 16mo.  Boards.  100  pp.  (Van  Nostrand  Science  Series  No. 
122.) 50  cents 

VAN  NOSTRAND'S  Year  Book  of  Mechanical  Engineering  Data. 

With  many  tables  and  diagrams.     (First  year  of  issue,  1910.).  .  .In  Press 

WALKER,  SYDNEY  F.     Steam  Boilers,  Engines,  and  Turbines. 

189  Illustrations.     8vo.     Cloth.     428  pp Net,  3 .00 

WATSON,  E.  P.  Small  Engines  and  Boilers.  A  manual  of  con- 
cise and  specific  directions  for  the  construction  of  small  steam  engines 
and  boilers  of  modern  types  from  five  horse-power  down  to  model 
sizes.  Illustrated.  12mo.  Cloth.  1 16  pp , 1 . 25 

ZEUNER,  A.  Technical  Thermodynamics.  Translated  from  the 
Fifth,  completely  revised  German  Edition  of  Dr.  Zeuner's  original  treat- 
ise on  thermodynamics,  by  J.  F.  Klein.  Illustrated.  8vo.  Cloth.  Two 
volumes Net,  8.00 

Any  book  in  this  list  will  be  sent  postpaid  to  any  address  in 
the  world  on  receipt  of  price,  by 

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